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Chapter 15: Problem 50

Write the equation for a sinusoidal wave propagating in the negative \(x\) -direction with a speed of \(120 . \mathrm{m} / \mathrm{s}\), if a particle in the medium in which the wave is moving is observed to swing back and forth through a \(6.00-\mathrm{cm}\) range in \(4.00 \mathrm{~s}\). Assume that \(t=0\) is taken to be the instant when the particle is at \(y=0\) and that the particle moves in the positive \(y\) -direction immediately after \(t=0\).

Short Answer

Expert verified
Question: A sinusoidal wave is observed to travel in the negative x-direction with a wave speed of 120 m/s. The oscillating particle is observed to swing back and forth through a 6.00 cm range and take 4.00 seconds to complete an oscillation. Find the equation of the wave. Answer: The equation of the wave is given by y(x,t) = 0.030 * cos(π/240 * x + 0.5π * t - π/2), where x is the position, and t is the time.

Step by step solution

01

Find the amplitude of the wave

The amplitude (A) is the maximum displacement of the particles from their mean position. In this problem, we are given that a particle is observed to swing back and forth through a 6.00 cm range. Therefore, the amplitude of the wave is half of this range. A = 6.00 cm / 2 = 3.00 cm = 0.030 m
02

Find the frequency and angular frequency of the wave

The time taken for a complete oscillation of the particle is given as 4.00 seconds. This time is called the time period (T) of the wave. The frequency (f) is the inverse of the time period. f = 1 / T = 1 / 4.00 s = 0.25 Hz The angular frequency (ω) is related to the frequency by the following formula: ω = 2πf ω = 2π(0.25 Hz) = 0.5π rad/s
03

Determine the wave number (k) of the wave

The wave speed (v) is given as 120 m/s. The wave number (k) is related to the wave speed and angular frequency by the following formula: v = ω / k Rearrange the equation and solve for k: k = ω / v = (0.5π rad/s) / (120 m/s) = π/240 rad/m
04

Write the equation for the sinusoidal wave

Since the wave is propagating in the negative x-direction and particles move in the positive y-direction immediately after t=0, the equation for the sinusoidal wave can be represented as: y(x,t) = A * cos(kx + ωt + φ) Here, φ is the phase constant. As we are assuming that at t=0, the particle is at y=0 and moves in the positive y-direction immediately after t=0, this means the wave is at its mean position and hence has a phase constant of -π/2. φ = -π/2 Now, substitute the values of A, k, ω, and φ in the equation: y(x,t) = 0.030 * cos(π/240 * x + 0.5π * t - π/2)

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Most popular questions from this chapter

Hiking in the mountains, you shout "hey," wait \(2.00 \mathrm{~s}\) and shout again. What is the distance between the sound waves you cause? If you hear the first echo after \(5.00 \mathrm{~s}\), what is the distance between you and the point where your voice hit a mountain?

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