Resonance and Velocity of Transverse Wave along a Stretched String

Resonance

When a body capable of vibration is displaced and then allowed to vibrate freely, it will vibrate with a frequency which is called the natural frequency. If an external periodic force is applied on the body and the body vibrates with the frequency of the force, the motion is called the forced vibration. Forced vibration leads to resonance. Resonance is the specific response of a system which is capable of vibrating with a certain frequency to an external force acting with the same frequency. For example, A suspension bridge has its own natural frequency. If the frequency of vibration coming out of marching soldier through the bridge is equal to the natural frequency of the bridge, then the bridge will vibrate violently with large amplitude and may collapse. That’s why the soldiers are ordered to break the steps while crossing the bridge.

The two conditions for occurring the resonance are

1. The frequency of applied force must be equal to the natural frequency of the system,
2. The applied force must be equal to the vibrating system.

The graph of the amplitude of vibration and frequency of forced vibration is shown below. The amplitude is large if the damping is small and vice versa.

Resonance air column tube: measurement of velocity of sound

The apparatus consists of a glass tube of length about one-meter and diameter 4 cm fitted on a vertical board with a meter scale attached to it. The tube is connected at its lower end by a rubber tube to reservoir which can be slided up and down. The tube and a part of the reservoir is filled with water. A tuning fork of known frequency is set into vibration and held horizontally above the mouth of the tube. The vibrating prongs of the tuning fork force the air in the tube to vibrate. As the vibration is forced, the frequency of tuning fork is same as the fundamental frequency of the pipe and the air inside the pipe is set into resonance by the periodic force. The length of air column in the tube is noted which is the first resonating length denoted by L₁. Let 'e' be the end correction and at the fundamental mode of vibration,

$$\begin{align*} L_1 + e &= \frac {\lambda }{4} \\ \text {or,}\: \lambda &= 4(L_1 + e) \dots (i) \\ \end{align*}$$

Now, the length of air column is increased till another loud sound is heard with the same tuning fork.This is called the second resonance and this corresponds to the first overtone. Length of air column for is resonance is three times the length of the first resonance. So,

$$\begin{align*} L_2 + e &= \frac {3\lambda }{4} \\ \text {or,} \: 3\lambda &= 4(L_2 + e) \dots (ii) \\ \text {Subtracting equation} \: (i)\: \text {from equation} \: (ii), \text {we have} \\ 2\lambda &= 4(L_2 – L_1) \\ \text {or,} \: \lambda &= 2(L_2 –L_1) \\ \text {Knowing the frequency of the tuning fork,} \\ \text { v can be calculated as} \\ v &= f\lambda = 2f(L_2 – L_1) \dots (iii) \\ \end{align*}$$

$$\begin{align*} L_1 + e &= \frac {\lambda }{4} = \frac {v}{4f} \\ \text {or,} L_1 &= \frac {v}{4f} – e \end{align*}$$ The velocity of sound can be calculated at 0^oC using the relation as $$\begin{align*} v_o &= v\sqrt {\frac {T_o}{T}} \\ &= v\sqrt {\frac {273}{273 + \theta }} \\ \end{align*}$$At STP, velocity of aound can be calculated with correction of humidity as $$\begin{align*} v_o &= \sqrt {\frac {P -0.35\times f}{p}}\sqrt {\frac {273}{273 + \theta }} \end{align*}$$where f is the aqueous tension or saturated vapour pressure of water at lab temperature \(\theta \).

Velocity of Transverse Wave along a Stretched String

Suppose a stretched string. Consider a single symmetric pulse moving from left to right along the string with speed, v as shown in the figure. For convenience, we can consider a frame of reference in which the pulse remains stationary and then, we run along the pulse, keeping it constantly in view. In this frame, the string appears to move from right to left with the speed v.

Consider a small segment AB of the pulse of length DL forming an arc of a circle of radius R and subtending an angle 2θ at the centre, O of the circle. A tension T in the string pulls tangentially on this segment at each end. The horizontal component Tcos θ at two ends of the pulse cancel each other while the vertical components add to form a radial restoring force F given by

$$\begin{align*} F &= 2(T\sin\theta ) 2T\theta = T2\theta \\ \text {where } \: \sin \theta = \theta \: \text {for small angle.} \\ \text {The small angle can be written as} \: 2\theta = \Delta L/R.\\ \text {If} \: \mu \: \text {is the mass per unit length or linear density of the string,} \\ \text { the mass of the segment is } \\ m &= \mu \:\Delta L \\ \text {Since the string segment is moving in an arc of a circle,} \\ \text { this force acts as a centripetal force } \\ \text {producing a centripetal acceleration towards the centre of the circle} \\ \text {given by} \\ a &= \frac {v^2}{R} \\ \text {So the centripetal force,} \\ F &= \frac {mv^2}{R} \\\text {or,} \: T2\theta &= \frac {\mu \Delta Lv^2}{R} \\ \text {or,} \: \frac {T\Delta L}{R} &= \frac {\mu \Delta Lv^2}{R} \\ \text {or,}\: T &= \mu \: v^2 \\ \text {or,} \: v &= \sqrt {\frac {T}{\mu }} \\ \end{align*}$$The frequency of the wave is fixed entirely by whatever generates the wave as,$$\begin{align*} v &= \lambda f \\ \text {or,} \: f &= \frac {v}{\lambda } = \frac {1}{\lambda } \sqrt {\frac {T}{\mu }} \\ \end{align*}$$