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Chapter 10: Problem 1512

Standing waves are produced by the superposition of two waves $y_{1}=0.05 \sin (3 \pi t-2 x)\( and \)y_{2}=0.05 \sin (3 \pi t+2 x)$ where \(\mathrm{x}\) and \(\mathrm{y}\) are in meters and \(\mathrm{t}\) is in seconds. The distance (in meters) between two consecutive nodes is (A) \(\pi / 2\) (B) \(\pi\) (C) \(0.5\) (D) \(1.0\)

Short Answer

Expert verified
The distance between two consecutive nodes in the standing wave produced by the given waves is \(\frac{\pi}{2}\) meters.

Step by step solution

01

Write down the given wave expressions

Given that two waves are: \(y_1 = 0.05\sin(3\pi t - 2x)\) and \(y_2 = 0.05\sin(3\pi t + 2x)\)
02

Find the resultant wave

The standing wave is formed by the superposition of two waves, so we need to add them: \(y = y_1 + y_2 = 0.05\sin(3\pi t - 2x) + 0.05\sin(3\pi t + 2x)\)
03

Simplify the combined wave equation

To simplify, we can use the trigonometric identity: \(\sin(a) + \sin(b) = 2\cos(\frac{a-b}{2})\sin(\frac{a+b}{2})\) In this case, \(a = 3\pi t - 2x\) and \(b = 3\pi t + 2x\). Using the identity: \(y = 0.05 [2\cos(\frac{(3\pi t - 2x) - (3\pi t + 2x)}{2})\sin(\frac{(3\pi t - 2x) + (3\pi t + 2x)}{2})]\) \(y = 0.1\cos(-2x)\sin(3\pi t)\)
04

Find the node positions

Nodes are points of zero amplitude. So, we need to find the values of x for which \(y = 0\). Setting the equation to zero: \(0 = 0.1\cos(-2x)\sin(3\pi t)\) As we are looking for the positions of nodes, we are concerned with the spatial term \(\cos(-2x)\) which becomes zero, and not the time-dependent term \(\sin(3\pi t)\). So, we have: \(\cos(-2x) = 0\)
05

Calculate the distance between consecutive nodes

The cosine term is zero when the argument is equal to odd multiples of \(\frac{\pi}{2}\), i.e., \(-2x = (2n + 1) \frac{\pi}{2}\) where \(n = 0, 1, 2, ...\) Solving for x, we have: \(x = -(2n + 1) \frac{\pi}{4}\) The distance between consecutive nodes is the difference between the positions of two consecutive nodes (when n and n+1): \[\begin{aligned} d &= x(n+1) - x(n) \\ &= - (2(n+1) + 1)\frac{\pi}{4} - (- (2n + 1)\frac{\pi}{4}) \\ &= - \frac{2\pi}{4} \\ &= - \frac{\pi}{2} \end{aligned}\] Since distance cannot be negative, we take the absolute value, so the distance between two consecutive nodes is \(\frac{\pi}{2}\) meters. Therefore, the correct answer is (A) \(\frac{\pi}{2}\).

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