Question:medium

A P-wave of frequency 20 Hz is travelling through a non-dispersive medium with a velocity of 5 km/s. The amplitude retained at a distance of 10 km from source is _____________ % (rounded off to one decimal place). (Use quality factor, \(Q = 80\))

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Use A/A0 = exp(-pi*f*x/(Q*v)); with f=20 Hz, x=10 km, v=5 km/s, Q=80 the exponent is pi/2, giving about 20.8% amplitude retained.
Updated On: Jul 21, 2026
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Correct Answer: 20

Solution and Explanation

An equivalent, and arguably more physically transparent, route is to work with the number of oscillation cycles the wave completes while travelling and the per-cycle amplitude decay factor, instead of the distance-attenuation formula directly.

First find the travel time from source to the 10 km observation point:

\[ t = \frac{x}{v} = \frac{10\ \text{km}}{5\ \text{km/s}} = 2\ \text{s} \]

The number of complete oscillation cycles the P-wave undergoes in this travel time is:

\[ N = f \times t = 20\ \text{Hz} \times 2\ \text{s} = 40\ \text{cycles} \]

The quality factor \(Q\) is defined through the fractional energy loss per radian of phase (per cycle, energy decays as \(e^{-2\pi/Q}\) per cycle, and since amplitude goes as the square root of energy, amplitude decays as \(e^{-\pi/Q}\) per cycle). Over \(N\) cycles, the total amplitude decay is:

\[ \frac{A}{A_0} = \left(e^{-\pi/Q}\right)^{N} = e^{-\pi N/Q} = e^{-\pi \times 40/80} = e^{-\pi/2} = e^{-1.5708} = 0.2079 \]

which is identical to \(\pi f x/(Qv)\) since \(N = ft = fx/v\) -- confirming the two approaches agree. As a percentage this is 20.8%, safely inside the official 20-21% band.

\(\boxed{\dfrac{A}{A_0} \approx 20.8\%}\)

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