24Stationary/ Standing waves

📚
STATIONARY / STANDING WAVES
▢ Stationary Wave:
❖ Definition: Result of superposition of two identical progressive waves of same frequency, same amplitude and same speed travelling in opposite directions
❖ Also Called: Standing wave
❖ Formation Condition:
  • •
    Two waves identical
  • •
    Same frequency
  • •
    Same amplitude
  • •
    Opposite directions
  • •
    Same medium
  • •
    Same speed
❖ Main Features:
  • •
    No net transfer of energy along the medium
  • •
    Energy is confined between nodes
  • •
    Amplitude varies from point to point
  • •
    Nodes and antinodes remain fixed
  • •
    Particles between two consecutive nodes vibrate in same phase
  • •
    Particles in adjacent loops vibrate in opposite phase
  • •
    Distance between consecutive nodes = λ/2
  • •
    Distance between consecutive antinodes = λ/2
  • •
    Distance between node and adjacent antinode = λ/4
▢ Nodes and Antinodes:

Table 1: Nodes vs Antinodes

Feature
Node
Antinode
Displacement
Minimum / zero
Maximum
Velocity
Minimum / zero
Maximum
Pressure
Maximum
Minimum
Energy
Maximum
Minimum
Amplitude
0
Maximum
Distance N-N
λ/2
-
Distance A-A
-
λ/2
Distance N-A
λ/4
λ/4
▢ Stationary Wave Equation:

Table 1: Equations

Case
Formula
Incident wave
y₁ = a sin(ωt − kx)
Reflected wave from free end
y₂ = a sin(ωt + kx)
Resultant at free end
y = 2a sinωt coskx
Amplitude at position x
A = 2a coskx
Reflected wave from rigid end
y₂ = a sin(ωt + kx + π)
Resultant at rigid end
y = −2a cosωt sinkx
Amplitude at position x
A = 2a sinkx
❖ Free Boundary:
  • •
    No phase reversal
  • •
    Antinode at free end
  • •
    Displacement maximum
  • •
    Pressure minimum
❖ Rigid Boundary:
  • •
    Phase reversal by π
  • •
    Node at rigid end
  • •
    Displacement minimum
  • •
    Pressure maximum
▢ Types of Stationary Waves:

Table 1: Types

Type
Medium / Example
Transverse stationary wave
Stretched string, sonometer wire
Longitudinal stationary wave
Air column in organ pipe, resonance tube
▢ Vibration of Stretched String:

Table 1: String Formulae

Quantity
Formula
Wave speed on string
v = √(T/μ)
Linear density
μ = m/l
For p loops
l = pλ/2
Wavelength
λ = 2l/p
Frequency
fₚ = p/(2l) √(T/μ)
Fundamental frequency
f₁ = 1/(2l) √(T/μ)
p-th harmonic
fₚ = p f₁
❖ Modes:

Table 1: String Modes

Mode
Loops
Frequency
Name
1st mode
1
f₁ = v/2l
Fundamental / 1st harmonic
2nd mode
2
f₂ = 2f₁ = v/l
1st overtone / 2nd harmonic
3rd mode
3
f₃ = 3f₁ = 3v/2l
2nd overtone / 3rd harmonic
❖ Harmonics: String fixed at both ends produces all harmonics: 1 : 2 : 3 : 4 : ...
▢ Mersenne's Laws:

Table 1: Laws of Transverse Vibration of String

Law
Condition
Relation
Law of length
T and μ constant
f ∝ 1/l
Law of tension
l and μ constant
f ∝ √T
Law of mass
l and T constant
f ∝ 1/√μ
Same material wire
l and T constant
f ∝ 1/D
Different material wire
l and T constant
f ∝ 1/(D√ρ)
❖ Combined Formula: f = p/(2l) √(T/μ)
❖ Wire Formula: f = p/(lD) √(T/πρ)
❖ Tension Formula: T = μ(2lf/p)²
▢ Sonometer:
❖ Definition: Instrument used to study vibration of stretched string and verify laws of vibrating string
❖ Principle: Resonance / unison between tuning fork and stretched wire

Table 1: Sonometer Relations

Condition
Relation
Same wire, same tension
f₁l₁ = f₂l₂
Same wire, same length
f₁/f₂ = √(T₁/T₂)
Same length and tension
f₁/f₂ = √(μ₂/μ₁)
Same material
f₁/f₂ = D₂/D₁
Same diameter
f₁/f₂ = √(ρ₂/ρ₁)
❖ Uses:
  • •
    Determine frequency of tuning fork
  • •
    Compare frequencies
  • •
    Verify laws of stretched string
  • •
    Study resonance
▢ Tuning Fork:

Table 1: Tuning Fork Facts

Point
Answer
Material
Elinvar
Property of Elinvar
Young's modulus does not change appreciably with temperature
Prongs
Transverse vibration
Stem
Longitudinal vibration
Sound
Single frequency
Loading with wax
Frequency decreases
Filing / cutting prongs
Frequency increases
Prongs vibration
Opposite phase
❖ Frequency: f = mt/(4√3 ρL²) √(Y/ρ)
❖ Symbols:
  • •
    m = constant
  • •
    t = thickness of prong
  • •
    L = length of prong
  • •
    Y = Young's modulus
  • •
    ρ = density
▢ Organ Pipe:
❖ Definition: Pipe containing air column in which longitudinal stationary waves are formed
❖ Types:
  • •
    Closed organ pipe
  • •
    Open organ pipe
❖ Important Points:
  • •
    Closed end → displacement node, pressure antinode
  • •
    Open end → displacement antinode, pressure node
  • •
    Air column resonance produces sound
  • •
    Frequency depends on effective length and velocity of sound
▢ Closed Organ Pipe:
❖ Definition: Pipe closed at one end and open at other
❖ Boundary Condition: Closed end = node; open end = antinode

Table 1: Closed Pipe Formulae

Mode
Length
Frequency
Name
1st mode
l = λ/4
f₁ = v/4l
Fundamental / 1st harmonic
2nd mode
l = 3λ/4
f₃ = 3v/4l
1st overtone / 3rd harmonic
3rd mode
l = 5λ/4
f₅ = 5v/4l
2nd overtone / 5th harmonic
p-th allowed mode
l = (2p−1)λ/4
f = (2p−1)v/4l
Odd harmonics only
❖ Harmonic Ratio: 1 : 3 : 5 : 7 : ...
❖ Missing Harmonics: Even harmonics absent
▢ Open Organ Pipe:
❖ Definition: Pipe open at both ends
❖ Boundary Condition: Both ends = antinodes

Table 1: Open Pipe Formulae

Mode
Length
Frequency
Name
1st mode
l = λ/2
f₁ = v/2l
Fundamental / 1st harmonic
2nd mode
l = λ
f₂ = v/l
1st overtone / 2nd harmonic
3rd mode
l = 3λ/2
f₃ = 3v/2l
2nd overtone / 3rd harmonic
p-th mode
l = pλ/2
fₚ = pv/2l
All harmonics
❖ Harmonic Ratio: 1 : 2 : 3 : 4 : ...
▢ Open vs Closed Pipe:

Table 1: Comparison

Feature
Closed pipe
Open pipe
Ends
One closed, one open
Both open
Fundamental frequency
v/4l
v/2l
Harmonics
Odd only
All
First overtone
3rd harmonic
2nd harmonic
Second overtone
5th harmonic
3rd harmonic
Same length relation
fopen = 2fclosed
fopen = 2fclosed
Same fundamental frequency
lclosed = lopen/2
lopen = 2lclosed
▢ End Correction:
❖ Definition: Distance by which displacement antinode is formed outside the open end of pipe

Table 1: End Correction Formulae

Quantity
Formula
End correction
e = 0.6r
Using diameter
e = 0.3D
Effective length of closed pipe
l + e
Effective length of open pipe
l + 2e
Closed pipe fundamental
f = v/[4(l+e)]
Open pipe fundamental
f = v/[2(l+2e)]
▢ Resonance Tube:
❖ Definition: Instrument used to determine velocity of sound in air using resonance of air column

Table 1: Resonance Tube Formulae

Quantity
Formula
First resonance length
l₁ + e = λ/4
Second resonance length
l₂ + e = 3λ/4
Wavelength
λ = 2(l₂ − l₁)
Velocity
v = 2f(l₂ − l₁)
End correction
e = (l₂ − 3l₁)/2
Maximum echo/air column condition
Resonance
▢ Kundt's Tube:
❖ Definition: Apparatus used to determine velocity of sound in gases, liquids and solids
❖ Principle: Stationary longitudinal waves in air/gas column indicated by dust heaps

Table 1: Kundt's Tube Facts

Fact
Answer
Dust heaps
Form at nodes
Distance between adjacent dust heaps
λ/2
Rod vibration
Longitudinal
Use
Velocity of sound in gas / solid
Frequency of rod
Usually same as frequency of air column at resonance
▢ Melde's Experiment:
❖ Aim: Verification of laws of transverse vibration of stretched string
❖ Principle: Resonance between tuning fork vibration and string vibration

Table 1: Melde's Law Results

Quantity
Relation
Frequency
f = p/(2l) √(T/μ)
Law of length
f ∝ 1/l
Law of tension
f ∝ √T
Law of mass
f ∝ 1/√μ
❖ Modes:
  • •
    String forms loops
  • •
    Loops are stationary wave segments
  • •
    Number of loops determines harmonic
  • •
    At resonance, amplitude becomes maximum
▢ Resonance:
❖ Definition: Special case of forced vibration in which amplitude becomes maximum when frequency of applied force equals natural frequency
❖ Condition: Forced frequency = natural frequency
❖ Examples:
  • •
    Sonometer wire with tuning fork
  • •
    Air column resonance tube
  • •
    Organ pipe
  • •
    Kundt's tube
  • •
    Bridge vibration by marching soldiers
❖ Important Points:
  • •
    Soldiers should not march in step on bridge
  • •
    At resonance, amplitude is maximum
  • •
    Energy transfer is maximum
  • •
    Resonance can be useful or destructive
▢ Loaded and Filed Tuning Fork:

Table 1: Frequency Change

Action
Effect on frequency
Loading prong with wax
Frequency decreases
Filing/cutting prong
Frequency increases
Increasing prong length
Frequency decreases
Decreasing prong length
Frequency increases
Increasing thickness
Frequency increases
Increasing density
Frequency decreases
▢ Temperature and Frequency:

Table 1: Temperature Effects

System
Effect
Tuning fork
Frequency decreases with rise in temperature
Organ pipe
Frequency generally increases with rise in temperature because velocity of sound increases
Sonometer wire
Frequency may change due to change in tension/length
Elinvar tuning fork
Frequency nearly unaffected by temperature
▢ Energy in Stationary Wave:

Table 1: Energy Distribution

Point
Energy / Pressure
Node
Pressure energy maximum
Antinode
Kinetic energy/displacement maximum; pressure minimum
During vibration
Energy oscillates between kinetic and potential form
Net transport
No net transfer of energy along wave
▢ Read and Digest:

Table 1: Important Points

Fact
Answer
Stationary wave
Superposition of two identical opposite progressive waves
Distance between node and antinode
λ/4
Distance between two consecutive nodes
λ/2
Distance between two consecutive antinodes
λ/2
Free end reflection
No phase change
Rigid end reflection
Phase reversal π
String fixed at both ends
Nodes at both ends
Open end of pipe
Displacement antinode, pressure node
Closed end of pipe
Displacement node, pressure antinode
Closed pipe
Odd harmonics only
Open pipe
All harmonics
Sonometer
Used to verify laws of stretched string
Resonance tube
Used to determine velocity of sound
Kundt's tube
Used to determine velocity of sound in gas/solid
Tuning fork material
Elinvar
Loading tuning fork
Frequency decreases
Filing tuning fork
Frequency increases
End correction
0.6r or 0.3D
Resonance
Maximum amplitude at forced frequency = natural frequency
▢ High-Yield Recall:

Table 1: Stationary Waves One-Liners

Fact
Answer
Stationary wave equation
y = 2a sinωt coskx
Rigid end equation
y = −2a cosωt sinkx
Node spacing
λ/2
Antinode spacing
λ/2
Node-antinode spacing
λ/4
String speed
v = √(T/μ)
String frequency
f = p/(2l)√(T/μ)
String fundamental
f₁ = v/2l
Open pipe fundamental
f₁ = v/2l
Closed pipe fundamental
f₁ = v/4l
Open pipe harmonics
1,2,3,4,...
Closed pipe harmonics
1,3,5,7,...
First overtone closed pipe
3rd harmonic
First overtone open pipe
2nd harmonic
End correction
e = 0.6r
Resonance tube velocity
v = 2f(l₂ − l₁)
Resonance tube end correction
e = (l₂ − 3l₁)/2
Tuning fork loading
Frequency ↓
Tuning fork filing
Frequency ↑
Sonometer relation
fl = constant
Law of tension
f ∝ √T
Law of mass
f ∝ 1/√μ
Law of length
f ∝ 1/l
Q1.
A wave in resonance tube is
📅CEE MBBS 2025
Q2.
If the length of closed and open organ pipe is same, then the ratio of fundamental frequencies is:
📅CEE MBBS 2024B
Q3.
In a stretched string, the ratio of consecutive frequencies of the harmonics is:
📅CEE MBBS 2023
Q4.
A string fixed at two ends has harmonic frequencies in the ratio:
📅CEE MBBS 2023
Q5.
In a resonance tube experiment using a 1000 Hz tuning fork, the fundamental resonance length is 0.175 m. Taking tube diameter as 3.32 cm, the speed of sound is:
📅CEE MBBS 2021
Q6.
What will be the ratio of fundamental frequencies of an open organ pipe and close organ pipe of same length?
📅CEE BDS 2024
Q7.
The first and second resonance are obtained at depth of 21.5 cm and 65 cm in a resonance air column experiment. The third resonance will be obtained at:
📅CEE BDS 2021
Q8.
In a resonance-column experiment, two successive resonant air-column lengths differ by \(31.5\,\mathrm{cm}\). Find the wavelength.
📅BP 2013
Q9.
A travelling wave completes 20 wavelengths over a distance of \(40\,\mathrm m\). Find its wavelength.
📅BP 2010
Q10.
A sonometer wire vibrating over \(50\,\mathrm{cm}\) produces \(800\,\mathrm{Hz}\). At unchanged tension and linear density, what vibrating length produces \(1000\,\mathrm{Hz}\)?
📅BP 2010
Q11.
A string of length \(0.4\,\mathrm m\) and mass \(10^{-2}\,\mathrm{kg}\) is fixed at both ends under tension \(1.6\,\mathrm N\). Identical pulses are launched from one end. What interval between pulses equals the round-trip travel time, allowing returning and incoming pulses to overlap?
📅BP 2009
Q12.
A sonometer wire vibrates at \(100\,\mathrm{Hz}\) with a load in air, \(80\,\mathrm{Hz}\) when the load is fully immersed in water, and \(60\,\mathrm{Hz}\) when immersed in another liquid. Neglect air buoyancy and find the relative density of that liquid.
📅BP 2009
Q13.
A string fixed at both ends vibrates in the lowest mode having an antinode one-quarter of its length from an end. Its frequency is \(100\,\mathrm{Hz}\). Find the next higher mode that again has an antinode at this point.
📅BP 2009
Q14.
What are notes whose frequencies are integral multiples of the fundamental frequency called?
📅MOE 2014
Q15.
A tuning fork is in unison with a \(60\,\mathrm{cm}\) sonometer wire. Increasing the vibrating length to \(70\,\mathrm{cm}\) produces 4 beats per second. Find the tuning fork frequency.
Q16.
The wave speed on a string is \(300\,\mathrm{m/s}\) and its frequency is \(1000\,\mathrm{Hz}\). Find the spacing between adjacent nodes of a standing wave.
Q17.
An air-filled pipe closed at one end has fundamental frequency \(512\,\mathrm{Hz}\). If the same pipe is opened at both ends, find its new fundamental frequency (neglect end corrections).
📅IOM 2010
Q18.
A string fixed at both ends vibrates at its second overtone. How many nodes and antinodes are present?
📅MOE 2013
Q19.
The first overtones of a closed pipe and an open pipe have equal frequencies. Find the ratio of their lengths \(L_c:L_o\).
📅MOE 2012, 2011
Q20.
Transverse waves travel at \(3000\,\mathrm{m/s}\) on a stretched wire. What is the speed if tension is quadrupled?
📅MOE 2012
Q21.
The first two resonant air-column lengths are \(21.5\,\mathrm{cm}\) and \(65.0\,\mathrm{cm}\). Find the third resonant length.
📅MOE 2011
Q22.
The timbre (quality) of a musical sound depends mainly on which feature?
Q23.
If tension in a string becomes four times its initial value, its transverse-wave speed becomes:
📅KU 2014
Q24.
A wave on a string is described by \(y=3\cos(100t-\pi x)\), where \(x\) is in centimetres. Find its wavelength. (The missing factor of \(\pi\) in the scanned equation has been restored.)
📅KU 2012
Q25.
A tuning fork of \(480\,\mathrm{Hz}\) drives a sonometer wire whose free fundamental frequency is \(240\,\mathrm{Hz}\). What is the frequency of forced vibration?
📅KU 2011
Q26.
A \(216\,\mathrm{Hz}\) tuning fork produces 8 beats per second with a string. When the string tension is increased slightly, the beat frequency decreases. Find the original string frequency.
📅KU 2010
Q27.
A standing wave between two fixed atoms has three displacement nodes (including the endpoints) and two antinodes. If the atoms are \(1.21\,\mathrm{\mathring A}\) apart, find the wavelength.
📅HIE 2011
Q28.
Successive resonances occur for air-column lengths \(0.16\,\mathrm m\) and \(0.50\,\mathrm m\). If sound speed is \(340\,\mathrm{m/s}\), find the tuning fork frequency.
📅BP 2009
Q29.
A resonating column of air contains
Q30.
Velocity of waves in a string depends upon:
Q31.
As air temperature increases, how does the fundamental frequency of an organ pipe change (length constant)?
Q32.
An open organ pipe has fundamental frequency \(f\). Half its length is immersed in water, making the remaining air column closed at the water surface. Neglect end corrections. Find the new fundamental frequency.
Q33.
In a stationary wave, nodes are the points having
Q34.
In a stationary wave antinodes are the points having
Q35.
At the open end of an organ pipe, the air-displacement pattern has:
Q36.
A closed organ pipe has fundamental frequency \(f\). Find the frequency of its first overtone.
Q37.
The displacement field is \(y=A\cos(2\pi nt)\cos(2\pi x/\lambda)\). What does it represent?
Q38.
As a vessel is filled with water, the length of the air column above the water decreases. How does the fundamental resonance frequency of this air column change?
Q39.
A pipe closed at one end has fundamental frequency \(512\,\mathrm{Hz}\). What is the fundamental frequency when its other end is opened (neglect end corrections)?
Q40.
A pipe open at both ends has fundamental frequency \(f\). If half the tube is dipped vertically into water, find the fundamental frequency of the remaining air column.
Q41.
For a circular organ pipe, the magnitude of end correction increases when the tube is:
Q42.
A pipe closed at one end has original length \(L\) and sound speed \(V\). Upon heating, its length increases by \(D\) and sound speed increases by \(v\). Find its new fundamental frequency.
Q43.
The end correction for a resonance column is \(e=1.0\,\mathrm{cm}\). Its first resonant physical air-column length is \(15.0\,\mathrm{cm}\). Find the next resonant length.
Q44.
The cross-sectional area of a wire is halved and its tension doubled, without changing vibrating length or material density. If its original fundamental frequency is \(f\), find the new frequency.
Q45.
A sonometer wire of length \(1.00\,\mathrm m\) vibrates at a fundamental frequency of \(330\,\mathrm{Hz}\). Find transverse-wave speed.
Q46.
Two wires of the same material and cross-sectional area have vibrating lengths \(L\) and \(2L\), with fundamental frequencies 100 Hz and 150 Hz respectively. Find \(T_1:T_2\).
Q47.
Equal-length wires of the same material have diameters in the ratio \(1:2\) and the same tension. Find their fundamental-frequency ratio.
Q48.
A sonometer wire has its length doubled and its tension quadrupled. Find the ratio of new to original fundamental frequency.
Q49.
A pipe open at both ends has fundamental frequency \(f\). It is dipped halfway into water. Find the fundamental frequency of its remaining closed air column.
Q50.
In a resonance-tube experiment, water is replaced by a denser, nonvolatile oil. If the air-column length and air temperature are unchanged, what happens to its resonant frequency?
📅MOE 2010/BP 2015
Q51.
For the standing wave \(y=\cos(60x)\sin(100\pi t)\), where \(x\) is in centimetres, find the smallest positive node coordinate.
📅IOM 07
Q52.
A closed pipe at its first overtone and an open pipe at its third harmonic resonate with the same tuning fork. Find \(L_{\mathrm{closed}}:L_{\mathrm{open}}\).
📅IOM 05
Q53.
Find the third harmonic of an open pipe of length \(50\,\mathrm{cm}\) when sound speed is \(332\,\mathrm{m/s}\).
📅MOE 066
Q54.
Successive air-column resonance lengths differ by \(31.5\,\mathrm{cm}\). What is the wavelength?
📅MOE 2065
Q55.
An open organ pipe of length \(27\,\mathrm{cm}\) and a closed pipe of length \(21\,\mathrm{cm}\) have identical first-overtone frequencies. Assuming the same correction \(e\) at each open end, find \(e\).
📅MOE 2061
Q56.
A \(1\,\mathrm m\) string of mass \(0.01\,\mathrm{kg}\) is under tension \(1000\,\mathrm N\). Find transverse-wave speed.
📅Bangladesh Emb
Q57.
An open pipe of length \(L_1\) and a closed pipe of length \(L_2\) have equal *fundamental* frequencies. Find \(L_1:L_2\).
📅MOE 09
Q58.
The fundamental frequency of a closed pipe equals the first overtone of an open pipe of length \(60\,\mathrm{cm}\). Find the closed-pipe length.
📅BPKIHS-95
Q59.
Find the *shortest* closed-pipe length that resonates with a sound wavelength of \(48\,\mathrm{cm}\).
📅BPKIHS-97
Q60.
Which standing-wave equation has a displacement antinode at the free end \(x=0\)?
Q61.
A stationary displacement wave is \(y=-2a\cos(\omega t)\cos(kx)\). What type of displacement point occurs at \(x=0\)?
Q62.
String A has twice the radius, material density, and tension of string B. Their lengths are equal. Find their fundamental-frequency ratio \(f_A:f_B\).
Q63.
A piano wire of diameter \(0.90\,\mathrm{mm}\) is replaced with wire of diameter \(0.93\,\mathrm{mm}\), all other properties unchanged. Find the approximate percentage change of fundamental frequency.
📅IOM/BPKIHS
Q64.
A \(200\,\mathrm{Hz}\) tuning fork is initially in unison with a sonometer wire. If its tension increases by 1%, what is the approximate beat frequency?
Q65.
A uniform string of original fundamental frequency \(n\) is cut into three pieces with fundamentals \(n_1,n_2,n_3\) under the same tension. Which relation is correct?
Q66.
If the tension in a sonometer wire decreases by \(19\%\), find the exact percentage change of frequency.
Q67.
A tuning fork gives 6 beats per second with a sonometer wire of vibrating length \(95\,\mathrm{cm}\) and also with one of length \(100\,\mathrm{cm}\), at unchanged tension and linear density. Find the tuning-fork frequency.
Q68.
A sonometer wire resonates with a \(256\,\mathrm{Hz}\) fork at length \(36\,\mathrm{cm}\). Find the fork frequency that resonates at \(48\,\mathrm{cm}\) with unchanged tension.
Q69.
At tension \(36\,\mathrm N\) a string driven at constant frequency forms six loops. Find the tension needed for four loops.
Q70.
A resonance column has end correction \(1.0\,\mathrm{cm}\) and shortest resonant air length \(15.0\,\mathrm{cm}\). Find the next resonant length.
Q71.
A vertical glass tube of length \(1\,\mathrm m\) is filled with water and then slowly drained while a \(500\,\mathrm{Hz}\) tuning fork is held above its open end. How many resonances occur if sound speed is \(330\,\mathrm{m/s}\)? Neglect end correction.
Q72.
A closed organ pipe of length \(1\,\mathrm m\) has sound speed \(320\,\mathrm{m/s}\). With which listed frequencies can it resonate?
Q73.
An open pipe has fundamental frequency \(300\,\mathrm{Hz}\). Its first overtone coincides with the first overtone of a closed pipe. If sound speed is \(320\,\mathrm{m/s}\), find the length of the closed pipe.
Q74.
Closing one end of an open organ pipe makes its first-overtone frequency \(100\,\mathrm{Hz}\) less than the original first overtone. Find the fundamental frequency of the original open pipe.
Q75.
For a standing wave \(y=5\sin\!\left(\frac{2\pi x}{3}\right)\cos(40\pi t)\) with \(x\) measured in centimetres, find the spacing of adjacent displacement nodes. (The wave-number term in the OCR source was incomplete; it has been reconstructed to match the keyed choice.)
Q76.
A sound wave travels at \(350\,\mathrm{m/s}\). Find the fundamental frequency of an open organ pipe of length \(50\,\mathrm{cm}\).
📅IOM 2017
Q77.
The first and second resonant air-column lengths are \(22.7\,\mathrm{cm}\) and \(70.2\,\mathrm{cm}\). Find the third resonant length.
Q78.
An open organ pipe of length \(40\,\mathrm{cm}\) has sound speed \(320\,\mathrm{m/s}\). A near-resonant driving frequency is \(1201\,\mathrm{Hz}\). Which overtone is nearest? (The precise resonant frequency is 1200 Hz.)