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.
If the difference in resonating lengths is 31.5 cm in resonance column. The wavelength produced is
📅BP 2013
Q2.
If 20 vibrations is produced in 40 m wire then the wavelength of the wave is
📅BP 2010
Q3.
A sonometer wire of 50 cm length produces 800 cycles per second. The length of sonometer wire required to produce 1000 cycles per second is:
📅BP 2010
Q4.
A string of length 0.4 m and mass 10⁻² kg is tightly clamped at its ends. The tension in the string is 1.6 N. Identical wave pulses are produced at one end at equal intervals of time Δt. The value of Δt which allows constructive interference between successive pulses is:
📅BP 2009
Q5.
The frequency of a sonometer wire is 100 Hz. When the weights producing the tensions are completely immersed in water, the frequency becomes 80 Hz and on immersing the weights in a certain liquid, the frequency becomes 60 Hz. The specific gravity of the liquid is:
📅BP 2009
Q6.
A string fixed at both ends is vibrating in the lowest mode of vibration for which a point at quarter of its length from one end is a point of maximum displacement. The frequency of vibration in this mode is 100 Hz. What will be the frequency emitted when it vibrates in the next mode such that this point is again a point of max. displacement?
📅BP 2009
Q7.
A group of notes which is integral multiple of fundamental note is called
📅MOE 2014
Q8.
A tuning fork is in unison with sonometer wire of 60 cm length. The length is increased by 10 cm then beat frequency becomes 4 beats/seThe frequency of the tuning fork is:
📅
Q9.
The sound travels with speed 300 m/s in string. Then find the distance between two successive nodes. If frequency is 1000 Hz
📅
Q10.
A tube closed at one end containing air when excited produces the fundamental note of frequency 512 Hz. If the tube is open at both ends the fundamental frequency that can be excited is
📅IOM 2010
Q11.
A sonometer wire is vibrating in its second overtone. There are
📅MOE 2013
Q12.
A closed organ pipe and an open organ pipe have their first overtone identical in frequency. Their lengths are in the ratio:
📅MOE 2012, 2011
Q13.
A transverse wave passes with a speed of 3000 m/s along a stretched wire. If the tension in the wire is increased four times, the velocity of wave will be:
📅MOE 2012
Q14.
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
📅MOE 2011
Q15.
Timber of music depends upon
📅
Q16.
In a string, if tension is increased by 4 times then the velocity of transverse wave in string increases by
📅KU 2014
Q17.
The equation of wave traveling in a string can be written as y = 3 cos(100t - x). Its wavelength is
📅KU 2012
Q18.
A tuning fork of frequency 480 Hz is used to vibrate a sonometer wire having natural frequency 240 Hz. The wire will vibrate with frequency of
📅KU 2011
Q19.
A tuning fork of 216 Hz is vibrating with a string then the beat frequency is 8 Hz. When the tension in the string is increased beat frequency decreases. Then frequency of the string must be:
📅KU 2010
Q20.
A standing wave between atoms has 3 nodes and 2 antinodes. The distance between the two atoms is 1.21 Å. The wavelength of the wave is:
📅HIE 2011
Q21.
A long glass tube is held vertically in water. A tuning fork is struck and held over the tube. Strong resonances are observed at two successive lengths 0.16 m and 0.50 m above the surface of water. If the velocity of sound is 340 m/s, then the frequency of the tuning fork is:
📅BP 2009
Q22.
A resonating column of air contains
📅
Q23.
Velocity of waves in a string depends upon:
📅
Q24.
With increase in temperature, the frequency of sound from an organ pipe
📅
Q25.
The fundamental frequency of an open organ pipe is f. If half of it is dipped into water, then new fundamental frequency will be
📅
Q26.
In a stationary wave, nodes are the points having
📅
Q27.
In a stationary wave antinodes are the points having
📅
Q28.
At open end of an organ pipe
📅
Q29.
The fundamental frequency of a closed organ pipe is f. The frequency of its first overtone is
📅
Q30.
The displacement is given by the equation y = A cos 2πnt cos(2πx/λ), where A, n, λ are constants. It represents
📅
Q31.
As an empty vessel is filled with water its frequency
📅
Q32.
A tube, closed at one end and containing air produces, when excited, the fundamental note of frequency 512 Hz. If the tube is open at both ends; the fundamental frequency that can be excited is (in Hz)
📅
Q33.
A cylindrical tube, open at both ends has fundamental frequency f in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of air column is now
📅
Q34.
The end correction for the vibration of air column in a tube of circular cross-section will be more if the tube is
📅
Q35.
A hollow metallic tube of length L and closed at one end produce resonance with tuning fork of frequency n. The entire tube is then heated carefully so that at equilibrium temperature its length changes by D. If the change in velocity V of sound is v, the resonance will now be produced by tuning fork of frequency
📅
Q36.
The end correction of a resonance column is 1.0 cm. If the shortest length resonating with the tuning fork is 15.0 cm, the next resonating length will be
📅
Q37.
The frequency of a vibrating wire is f. When area of cross section of a wire is halved and tension doubled, the frequency becomes
📅
Q38.
A sonometer wire, 100 cm in length, has a fundamental frequency of 330 Hz. The velocity of propagation of transverse waves along this wire is
📅
Q39.
Two stretched wires of same material of lengths l and 2l vibrate with frequencies 100 and 150 Hz respectively. The ratio of their tension is
📅
Q40.
Two wires made of the same material are of equal lengths but their diameters are in the ratio of 1:2. On stretching each of these two strings by same tension, the ratio between the fundamental frequencies of these strings is
📅
Q41.
The length of a sonometer wire is doubled and its tension is increased four times. The fundamental frequency of the wire is changed in the ratio
📅
Q42.
A cylindrical tube, open at both ends has a fundamental frequency f in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of air column is now:
📅
Q43.
If oil of density higher than water is filled in place of water in a resonance tube, its frequency will
📅MOE 2010/BP 2015
Q44.
In the equation y = cos(60x)sin(100πt), x and y are in cm and t is in seconds. At the node find the value of x.
📅IOM 07
Q45.
An organ pipe P₁, closed at one end and vibrating in its first overtone and another pipe P₂, open at both ends are vibrating in its third harmonic are in resonance with a given tuning fork. The ratio of the length of P₁ and P₂ is:
📅IOM 05
Q46.
The third harmonic of open ended pipe of length 50 cm is
📅MOE 066
Q47.
In a resonance tube the air columns for the first and second resonance differ by 31.5 cm. The wavelength of the sound waves in the tube is
📅MOE 2065
Q48.
One open organ pipe of l = 27 cm and closed organ pipe of length 21 cm sound in unison in their 1st overtone. Calculate the end correction for both pipes.
📅MOE 2061
Q49.
A string has mass 0.01 kg and has length 1 m. If the tension is 1000 N, the velocity of transverse wave in the string is
📅Bangladesh Emb
Q50.
An open pipe of length L₁ and closed pipe of length L₂ resonate to the same tuning fork. The ratio of their lengths (L₁/L₂) is
📅MOE 09
Q51.
The fundamental frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. If the length of the open pipe organ is 60 cm, what is the length of the closed pipe?
📅BPKIHS-95
Q52.
Resonance will be produced with sound waves of 48 cm, in a closed pipe of length
📅BPKIHS-97
Q53.
Identify the equation of stationary wave for free end:
📅
Q54.
The equation of stationary wave is given by y = -2a cosωt cos kx. The phase difference between the incident wave and the reflected wave is
📅
Q55.
The radius, density and tension of string A is twice the radius, density and tension of another string B. If the length of both strings are equal, then the ratio of their frequency of vibration is:
📅
Q56.
A piano wire of diameter 0.9 mm is replaced by another wire of 0.93 mm. Then the percentage change in frequency of piano wire is
📅IOM/BPKIHS
Q57.
A tuning fork of 200 Hz is in unison with sonometer wire. If percentage increase in tension of wire is 1%, then number of beats produced per second is
📅
Q58.
If n₁, n₂, n₃ are the fundamental frequencies of three segments into which a string is divided. The original fundamental frequency of wire is
📅
Q59.
If tension in sonometer wire decreases by 19%, then percentage change in frequency is
📅
Q60.
A sonometer string and a tuning fork when sounded together give 6 beats/sec whether length of sonometer string is 95 cm or 100 cm. The frequency of tuning fork is
📅
Q61.
A string of 36 cm length was in unison with a fork of frequency 256 per second. It was in unison with another fork when the vibrating length was 48 cm, the tension being unchanged. The frequency of the second fork is
📅
Q62.
A cord attached to a vibrating tuning fork is divided into six segments under a tension of 36 N. It will be divided into 4 segments if the tension is
📅
Q63.
In a resonance column, first and second resonance are obtained at depths 22.7 cm and 70.2 cm respectively. The third resonance will be obtained at a depth
Q64.
The end correction of a resonance column is 1.0cm. If the shortest length resonating "with the tuning fork is 15.0cm, the next resonating length will be
Q65.
A glass tube Im length is filled by water. The water can be drained out slowly at the bottom of the tube. If a vibrating tuning fork of frequency 500c/s is brought at the upper end of the tube and the velocity of sound is 330m/s, then the total number of resonance obtained will be
Q66.
The speed of sound in air is 320m/s. A closed organ pipe of length Im can resonate with a frequency of
Q67.
An open organ pipe has fundamental frequency 300 Hz. The frequency of first overtone of open organ pipe is equal to the frequency of first overtone of closed organ "pipe. If speed of sound in air is 320m/s then length of closed organ pipe is
Q68.
An open organ pipe is suddenly closed at one end with the resultant frequency of first overtone of open organ pipe is 100 Hz more than frequency of first overtone of closed . organ pipe. The fundamental frequency of open organ pipe is
Q69.
The equation of a stationary wave is: y = 5sin 3 cos40nt Where x and y are in cm and t in seconds. Then the separation between two consecutive nodes is:
Q70.
The velocity of sound is 350 m/s. The length of open organ pipe is 50.cm. Find its fundamental frequency. . [IOM 2017]
📅IOM 2017