Question

A sinusoidal wave is sent along a cord under tension, transporting energy at the average rate of${\mathit{P}}_{\mathbf{avg}\mathbf{1}}$,1.Two waves, identical to that first one, are then to be sent along the cord with a phase difference $\phi$of either0,0.2 wavelength, or 0.5wavelength. (a) With only mental calculation, rank those choices of$\phi$according to the average rate at which the waves will transport energy, greatest first. (b) For

The first choice of$\phi$, what is the average rate in terms oflocalid="1661229908063" ${\mathit{P}}_{\mathbf{avg}\mathbf{1}}$?

Short answer

a) The rank of the phase difference is 0 wave length, 0.2 wave length, 0.5 wave length(zero).

b) The value of the greatest average rate of energy transfer in terms$P_{\mathrm{avg}1}$ of is$P_{avg}=4\times P_{\mathrm{avg}1}$

Step-by-step solution

Given

1. The average rate of energy transport is,$P_{\mathrm{avg}1}$.
2. The phase difference of the wavelengths is 0, 0.2 and 0.5 wave lengths.

Determining the concept

According to the equation of rate of energy transfer, the rate depends on the amplitude of the resultant wave. So, by knowing the amplitude of the resulting wave, rank the phase constants.

Formulae are as follow:

$P_{\mathrm{avg}1}=\frac{1}{2}\mu v{w}^2{y}_m^2$

$y_m=2y_m\cos \left(\frac{1}{2}\phi \right)$

Where, P is power, is phase difference, v is wave speed, ,𝝁 is mass per unit length,𝝎 is angular frequency

(a) Determining the rank of phase according to the average rate of transporting energy

The amplitude of the resultant of two interfering waves is given by,

$y_m=2y_m\cos \left(\frac{1}{2}\phi \right)$

Using this equation, determine (just by mental calculation) that the amplitude of the wave having phase difference 0 has maximum wavelength while that of 0.5 has minimum wavelength, and at amplitude 0.2 wavelength of the phase, the difference is intermediate.

The equation of power also implies that ,

$P_{avg1}\propto y_m^2$

So, rank the phase difference as,

1. Wave length, 0.2 wave length, 0.5 wave length.

Hence, the rank of the phase difference is 0 wave length, 0.2 wavelength, 0.5 wavelength.

(b) determining the value of the greatest average rate of energy transfer in terms ofPavg1.

Now, for the first choice the average rate is ,

$P_{\mathrm{avg}}=\frac{1}{2}\mu v{w}^2{y}_m^2$

.

$P_{\mathrm{avg}1}=4\times \frac{1}{2}\mu v{w}^2{y}_m^2$

$P_{\mathrm{avg}}=4\times P_{\mathrm{avg}1}$

Hence, the value of the greatest average rate of energy transfer in terms of$P_{\mathrm{avg}1}$ is$P_{\mathrm{avg}}=4\times P_{\mathrm{avg}1}$

Therefore, rank the phase differences using the equation of amplitude of the resultant wave. To find the average rate of energy transport, use the equation of power.