Two wave pulses are generated in a string. One of the pulses...

Two wave pulses are generated in a string. One of the pulses is given by equation $y_{1}=A \sin (\omega t-k x)$. If average power transmitted by both the pulses along the string are same and is given by $P=\frac{T A^{2} \omega^{2}}{2 v}$, where $T$ is the tension in the string, $A$ is amplitude of a pulse, $\omega$ is angular frequency of the source, and $v$ is wave velocity, then which one of the following equations may represent the other wave pulse? (A) $y_{2}=\frac{A}{\sqrt{2}} \sin (2 \omega t-k x)$ (B) $y_{2}=\frac{A}{\sqrt{2}} \sin (\omega t-2 k x)$ (C) $y_{2}=2 A \sin \left(\frac{\omega t}{2}-k x\right)$ (D) $y_{2}=2 A \sin \left(\frac{\omega t}{2}-\frac{k x}{2}\right)$

Two wave pulses are generated in a string. One of the pulses is given by equation $y_{1}=A \sin (\omega t-k x)$. If average power transmitted by both the pulses along the string are same and is given by $P=\frac{T A^{2} \omega^{2}}{2 v}$, where $T$ is the tension in the string, $A$ is amplitude of a pulse, $\omega$ is angular frequency of the source, and $v$ is wave velocity, then which one of the following equations may represent the other wave pulse?
(A) $y_{2}=\frac{A}{\sqrt{2}} \sin (2 \omega t-k x)$
(B) $y_{2}=\frac{A}{\sqrt{2}} \sin (\omega t-2 k x)$
(C) $y_{2}=2 A \sin \left(\frac{\omega t}{2}-k x\right)$
(D) $y_{2}=2 A \sin \left(\frac{\omega t}{2}-\frac{k x}{2}\right)$

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Key Concepts

Amplitude-Frequency Trade-off in Constant Power Transmission

When two waves are required to transmit the same average power, adjustments in amplitude must be balanced by corresponding changes in angular frequency (or vice versa) such that the product A²?² remains constant. This concept is essential when comparing or designing waves with differing parameters while needing to maintain equivalent energy transmission.

Sinusoidal Wave Representation

A sinusoidal wave is generally expressed as y = A sin(?t ? kx), where A represents the amplitude, ? is the angular frequency, and k is the wave number. This form is fundamental for describing periodic disturbances in a medium, encoding the wave’s spatial and temporal oscillations.

Average Power in Wave Motion

For waves on a string, the average power transmitted is given by an equation that depends critically on the string’s tension, as well as the amplitude and angular frequency of the wave, typically in the form P = (T A² ?²)/(2v). This relationship shows that the power increases with both the square of the amplitude and the square of the frequency, meaning changes in these parameters have a significant impact on energy transmission.

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