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Chapter 10: Problem 1518

The equation \(\mathrm{y}=10 \sin (\pi \mathrm{x} / 4) \cos 10 \pi t\) represents a stationary wave where \(\mathrm{x}\) and \(\mathrm{y}\) are in centimeter and \(\mathrm{t}\) is in seconds. The amplitude of each component wave is \(\ldots \ldots .\) (A) \(5 \mathrm{~cm}\) (B) \(10 \mathrm{~cm}\) (C) \(20 \mathrm{~cm}\) (D) between \(5 \mathrm{~cm}\) and \(10 \mathrm{~cm} .\)

Short Answer

Expert verified
The amplitude of each component wave is 10 cm. Therefore, the correct answer is (B) \(10 \mathrm{~cm}\).

Step by step solution

01

Identify the given stationary wave equation

The given stationary wave equation is \(y = 10 \sin(\pi x / 4) \cos(10\pi t)\).
02

Identify the component wave functions

The equation is a product of sine and cosine functions. The functions are: 1. \(\sin(\pi x / 4)\) 2. \(\cos(10 \pi t)\)
03

Determine the amplitude of each component wave

Each component wave's amplitude is the maximum value of the corresponding sine or cosine function. For both sine and cosine functions, the maximum value is 1. Therefore, we need to find the coefficient that multiplies each function to determine the amplitude. For the first component wave (the sine function), the coefficient is 10. The amplitude of this component wave is: \(A_1 = 10(1) = 10\mathrm{~cm}\) For the second component wave (the cosine function), the coefficient is also 10, but it is affected by the first component wave function. The amplitude of this component wave is: \(A_2 = 10(1) = 10\mathrm{~cm}\)
04

Determine the correct answer

The amplitude of each component wave is 10 cm. Therefore, the correct answer is (B) \(10 \mathrm{~cm}\).

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