Question

A transverse wave of amplitude 0.50 mm and frequency 100 Hz is produced on a wire stretched to a tension of 100 N. If the wave speed is 100 m/s. What average power is the source transmitting to the wire?

Answer 1

Hint: When the wave or transverse wave propagates through a medium, it disturbs and transmits power or loss power. So the average power loss can be calculated using the relation between the power, frequency, amplitude, tension and mass density. To calculate the mass density, we will use the formula of velocity which is given as \[V=\sqrt{\dfrac{T}{u}}.\]

Formula Used:
The formula of average power loss is given as,
\[P=2{{\pi }^{2}}uV{{A}^{2}}{{f}^{2}}\]
where u = mass density (linear), v = speed, A = amplitude, f = frequency.

Complete step by step answer:
We have been given a transverse wave having the amplitude 0.5 mm and frequency of 100 Hz which is produced when the wire gets stretched to a tension of 100 N.
The speed of the wave (V) = 100 m/s.
Now, we need to calculate the average power when the source is transmitting to the wire. We know that the formula of average power is given by
\[{{P}_{avg}}=2{{\pi }^{2}}uV.{{A}^{2}}{{f}^{2}}.......\left( i \right)\]
where u = mass density, v = velocity of wave, A = amplitude, f = frequency.
Now, we have given the values of
\[\text{Amplitude}\left( A \right)=0.5mm=5\times {{10}^{-4}}m\]
Frequency = 100 Hz
Tension = 100 N
Speed of the wave = 100 m/sec
We are not given the value of the linear mass density u. Let us calculate the value of u.
We know that the formula of velocity is given as,
\[v=\sqrt{\dfrac{T}{u}}\]
\[\Rightarrow u=\dfrac{T}{{{v}^{2}}}\]
\[\Rightarrow u=\dfrac{100}{100\times 100}={{10}^{-2}}kg/m\]
Put all the values in equation (i), we get,
\[{{P}_{avg}}=2\times {{\left( 0.14 \right)}^{2}}\times {{10}^{-2}}\times 100\times 25\times {{10}^{-8}}\times {{\left( {{10}^{2}} \right)}^{2}}\]
\[\Rightarrow {{P}_{av}}=49\times {{10}^{-3}}\text{ Watt}\]
\[\Rightarrow {{P}_{av}}=49\text{ mWatt}\]
So this is the power that is delivered to the wave by the source.

Additional Information:
The speed of mechanical wave or transverse wave can be determined by the restoring force setup in the medium when it is disturbed by an external thing and the inertial properties also called as mass density of the medium. The speed of the wave is directly related to the former and inversely to the latter. For the wave created on the string, a restoring force is provided by the tension T in the string. In this case, the inertial property will be linear mass density u. The dimension of the quantity \[\dfrac{T}{u}\] is \[\left[ {{L}^{2}}{{T}^{-2}} \right].\]

Note:
The speed of the wave (v) depends only on the properties of the medium where it is propagating, i.e. T and \[{{u}_{o}}.\] T is a property of stretched string that arises due to an external force. The speed does not depend upon the wavelength of the wave \[\left( \lambda \right)\] and the frequency of the wave itself. The waves form on the string has a sinusoidal form.