# Wave in Pipes and Strings

## Wave in Pipes and String

1. Closed Pipe
The pipe having it’s one end open and it’s other end closed is called closed pipe. Example: resonance tube.
Let us consider a closed pipe PQ of length ‘l’ is sets into resonance condition by putting a vibrating turning fork near to it’s open end. Let ‘v’ be the velocity of sound wave in air. Then the frequencies for different modes of vibration can be calculated as below:

• Fundamental Mode of Vibration

In this mode of vibrations, there lies one node and one antinode between closed and open end of the pipe. If $$\lambda_{1}$$ be the wavelength of wave for this mode of vibration then,
$$l= \frac{\lambda_{1}}{4}$$
$$or, \lambda_{1} = 4l$$
​​​​​​
Then,
$$f_{1}= \frac{v}{\lambda_{1}}$$
$$f_{1}= \frac{v}{4l}$$
This is the frequency for fundamental mode of vibration called fundamental frequency or first harmonic.

• Second Mode of Vibration

In this mode of vibrations, there lies two node and two antinode between closed and open end of the pipe. If $$\lambda_{2}$$ be the wavelength of wave for this mode of vibration then,
$$l= \frac{3 \lambda_{2}}{4}$$
$$or, \lambda_{2} = \frac{4l}{3}$$
​​​​​​
Then,
$$f_{2}= \frac{v}{\lambda_{2}}$$
$$or, f_{2}= \frac{3v}{4l}$$
$$or, f_{2}= 3 f_{1}$$
This is the frequency for second mode of vibration or third harmonic or first overtone.

• Third Mode of Vibration

In this mode of vibrations, there lies three node and three antinode between closed and open end of the pipe. If $$\lambda_{3}$$ be the wavelength of wave for this mode of vibration then,
$$l= \frac{5 \lambda_{3}}{4}$$
$$or, \lambda_{3} = \frac{4l}{5}$$
​​​​​​
Then,
$$f_{3}= \frac{v}{\lambda_{2}}$$
$$or, f_{3}= \frac{5v}{4l}$$
$$or, f_{3}= 5 f_{1}$$
This is the frequency for third mode of vibration or third harmonic or second overtone.

In the similar way, frequency for nth mode of vibration is ,
$$f_{n}= (2n-1)f_{1}$$

2. Open Pipe
The pipe having it's both end's open is called open pipe.
Let us consider a open pipe PQ of length ‘l’ is sets into resonance condition by putting a vibrating turning fork near to it’s open end. Let ‘v’ be the velocity of sound wave in air. Then the frequencies for different modes of vibration can be calculated as below:
• Fundamental Mode of Vibration

In this mode of vibrations, there lies one node and two antinode between two open ends of the pipe. If $$\lambda_{1}$$ be the wavelength of wave for this mode of vibration then,
$$l= \frac{\lambda_{1}}{2}$$
$$or, \lambda_{1} = 2l$$
​​​​​​
Then,
$$f_{1}= \frac{v}{\lambda_{1}}$$
$$f_{1}= \frac{v}{2l}$$
This is the frequency for fundamental mode of vibration called fundamental frequency or first harmonic.

• Second Mode of Vibration

In this mode of vibrations, there lies two node and three antinode between two open ends of the pipe. If $$\lambda_{2}$$ be the wavelength of wave for this mode of vibration then,
$$l= \lambda_{2}$$
$$or, \lambda_{2} = l$$
​​​​​​
Then,
$$f_{2}= \frac{v}{\lambda_{2}}$$
$$or, f_{2}= \frac{v}{l}$$
$$or, f_{2}= 2 f_{1}$$
This is the frequency for second mode of vibration or second harmonic or first overtone.

• Third Mode of Vibration

In this mode of vibrations, there lies three node and four antinode between two open ends of the pipe. If $$\lambda_{3}$$ be the wavelength of wave for this mode of vibration then,
$$l= \frac{3 \lambda_{3}}{2}$$
$$or, \lambda_{3} = \frac{2l}{3}$$
​​​​​​
Then,
$$f_{3}= \frac{v}{\lambda_{2}}$$
$$or, f_{3}= \frac{3v}{2l}$$
$$or, f_{3}= 3 f_{1}$$
This is the frequency for third mode of vibration or third harmonic or second overtone.

In the similar way, frequency for nth mode of vibration is ,
$$f_{n}= n .f_{1}$$

End Correction
The distance from open end of the pipe to the point where antinode exactly forms is called end correction. It is generally denoted by the symbol ‘e’ and it’s value was calculated by Ray leigh i.e 0.3 d, where d is the internal diameter of the pipe. For closed pipe; e = 0.3 d whereas for open pipe e= 0.6 d.

Measurement of velocity of sound in air

The resonance apparatus to measure the velocity of sound in air is shown in the above figure. This experiment is based on the principle of resonance. The apparatus consist of a closed pipe of 1m length and diameter 5cm. Through the closed end of the pipe, a rubber tube is connected to a water reservoir. The level of water inside the pipe can be changed by changing the position of reservoir. The pipe is fixed vertically onto the wooden board with a meter scale to measure the length of the air column into the pipe.

Now, by putting a vibrating tuning fork near to the open end of the pipe, the position of 1st and 2nd resonance is measured. If l1 and l2 be the first and second resonating length respectively, λ be the wave length of wave and ‘e’ be the end correction of pipe then,

At first resonance,
$$l_{1} + e = \frac{\lambda}{4}............(i)$$

At second resonance,
$$l_{2} + e = \frac{3 \lambda}{4}............(ii)$$

Subtracting eqn (i) from (ii),
$$l_{2} - l_{1} = \frac{3 \lambda}{4} - \frac{\lambda}{4}$$
$$or, \frac{\lambda}{2} = l_{2} - l_{1}$$
$$or, \lambda = 2(l_{2} - l_{1})$$

If ‘v’ be the velocity of sound at room temperature then,

$$v = f \times \lambda$$
$$or, v = 2f(l_{2} - l_{1})............(iii)$$
where, f= frequency of tuning fork

Let v0 be the velocity of sound at 0oC, then
$$\frac{v}{v_{o}} = \sqrt{\frac{\theta + 273}{273}}$$
$$v_{o} = v. \sqrt{\frac{273}{\theta + 273}}$$

In this way velocity of sound can be determined by using resonance tube experiment.

Velocity of Transverse Wave in Stretched String
In a stretched string the velocity of a transverse wave depends upon tension acting on the wire ‘T’ as well as mass per unit length of the wire (μ),

$$i.e. v \propto T^{a} \mu^{b}$$
where a and b sre the constant whose value is to be determined.
$$or, v= k T^{a} \mu^{b}.......(i)$$
where k is dimensionless constant.

Using the dimensional formula of corresponding physical quantities in eqn (i),
$$or, [LT^{-1}] = [MLT^{-2}]^{a}[ML^{-1}]^{b}$$
$$or, [LT^{-1}] = [M^{a+b}L^{a-b}T^{-2a}]$$

Equating the corresponding elements,
$$a+b=0$$
$$or, a = -b ............(ii)$$

$$a-b= 1$$
$$or, -b-b= 1$$
$$or, -2b= 1$$
$$or, b= \frac{-1}{2}............(iii)$$

Substuting the value of a and b in eqn (i),
$$or, v= k T^{\frac{1}{2}} \mu^{\frac{-1}{2}}$$
$$or, v = k \sqrt{\frac{T}{\mu}}$$
Since k=1
$$or, v = \sqrt{\frac{T}{\mu}}$$

This is the expression for velocity .

Frequencies for different modes of vibration in strings

Let us consider a string PQ of length ‘l’ is kept stretched in between two rigid supports. It is allowed to vibrate into resonance condition by putting a vibrating turning fork near to it. Let ‘v’ be the velocity of sound wave in air. Then the frequencies for different modes of vibration can be calculated as below:

• Fundamental Mode of Vibration

In this mode of vibrations, there lies one node and two antinode in between two fixed end of the string. If $$\lambda_{1}$$ be the wavelength of wave for this mode of vibration then,
$$l= \frac{\lambda_{1}}{2}$$
$$or, \lambda_{1} = 2l$$
​​​​​​
Then,
$$f_{1}= \frac{v}{\lambda_{1}}$$
$$f_{1}= \frac{v}{2l}$$
This is the frequency for fundamental mode of vibration called fundamental frequency or first harmonic.

• Second Mode of Vibration

In this mode of vibrations, there lies two node and three antinode in between two fixed end of the string. If $$\lambda_{2}$$ be the wavelength of wave for this mode of vibration then,
$$l= \lambda_{2}$$
$$or, \lambda_{2} = l$$
​​​​​​
Then,
$$f_{2}= \frac{v}{\lambda_{2}}$$
$$or, f_{2}= \frac{v}{l}$$
$$or, f_{2}= 2 f_{1}$$
This is the frequency for second mode of vibration or second harmonic or first overtone.
• Third Mode of Vibration

In this mode of vibrations, there lies three node and four antinode in between two fixed end of the string. If $$\lambda_{3}$$ be the wavelength of wave for this mode of vibration then,
$$l= \frac{3 \lambda_{3}}{2}$$
$$or, \lambda_{3} = \frac{2l}{3}$$
​​​​​​
Then,
$$f_{3}= \frac{v}{\lambda_{2}}$$
$$or, f_{3}= \frac{3v}{2l}$$
$$or, f_{3}= 3 f_{1}$$
This is the frequency for third mode of vibration or third harmonic or second overtone.

In the similar way, frequency for nth mode of vibration is ,
$$f_{n}= n .f_{1}$$

Laws of transverse vibration in a stretched string
The laws of transverse wave of vibration in stretched string state that the frequency of transverse wave in stretched string is,

• Directly proportional to the square root of tension acting on the wire
$$i.e f \propto \sqrt{T}$$
• Inversely proportional to the square root of mass per unit length of the wire
$$i.e f \propto \frac{1}{\sqrt{\mu}}$$
• Inversely proportional to the resonating length of the wire
$$i.e f \propto \frac{1}{l}$$

Combining eqn (i), (ii) and (iii) we get,
$$f \propto \frac{1}{l} \sqrt{\frac{T}{\mu}}$$

Experimental Verification
To verify these laws, a sonometer is used. It consists of a hollow wooden box with two side holes. A wire is passed over the board. One end of the wire is fixed at a hook and the other end passes over a movable pulley through which a pan is suspended to put the variable mass. There a two wooden bridges which are moveable onto the board. There is a scale fixed over the board to measure the resonating length of the wire.

1. To Verify $$f \propto \sqrt{T}$$
For the verification of this law; mass per unit length of the wire ‘μ’ and resonating length of wire ‘l’ are kept constant . Using different value of tension on the wire, corresponding frequencies are measured when the wire vibrates in resonance condition. A graph of f and $$\sqrt{T}$$ is a straight line passing through the origin. Hence this law is verified.
2. To Verify $$f \propto \frac{1}{\sqrt{\mu}}$$
For the verification of this law; tension acting on the wire ‘T’ and resonating length of wire ‘l’ are kept constant . Using wires having different value of μ, corresponding frequencies are measured when the wire vibrates in resonance condition. A graph of f and $$\frac{1}{\sqrt{\mu}}$$ is a straight line passing through the origin. Hence this law is verified.

3. To Verify $$f \propto \frac{1}{l}$$
For the verification of this law; tension acting on the wire ‘T’ and mass per unit length of the wire ‘μ’ are kept constant . Using wires having different value of resonating length ‘l’ , corresponding frequencies are measured when the wire vibrates in resonance condition. A graph of f and  $$\frac{1}{l}$$is a straight line passing through the origin. Hence this law is verified.

Credit : Prof. Hari Saran Regmi