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Chapter 13: Problem 61

A wave of amplitude \(0.30 \mathrm{~m}\) interferes with a second wave of amplitude \(0.20 \mathrm{~m}\) traveling in the same direction. What are (a) the largest and (b) the smallest resultant amplitudes that can occur, and under what conditions will these maxima and minima arise?

Short Answer

Expert verified
The largest resultant amplitude that can occur is \(0.50m\) under the condition of constructive interference, while the smallest resultant amplitude that can occur is \(0.10m\) under the condition of destructive interference.

Step by step solution

01

Understanding the Problem

We have two wave amplitudes which are going to interfere with each other. One with an amplitude of 0.30m and the other with amplitude of 0.20m
02

Calculate Maximum Amplitude

The maximum amplitude, or the constructive interference, occurs when the two waves are perfectly in sync. This is when they reach their maximum amplitude at the same time and their amplitudes simply add together. So, maximum amplitude= amplitude of wave 1 + amplitude of wave 2 = \(0.30m + 0.20m = 0.50m\)
03

Calculate Minimum Amplitude

The minimum amplitude, or the destructive interference, occurs when the two waves are perfectly out of sync. This is when one wave reaches its maximum amplitude while the other is at its minimum. In this case, the amplitudes subtract. So, minimum amplitude = amplitude of wave 1 - amplitude of wave 2 = \(0.30m - 0.20m = 0.10m\)
04

Understand the Conditions for Maxima and Minima

The constructive (maximal) interference happens when the waves are in phase, that is, when the crests of the two waves coincide. The destructive (minimal) interference occurs when the waves are out of phase, that is, when the crest of one wave coincides with the trough of the other.

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Most popular questions from this chapter

A spring of negligible mass stretches \(3.00 \mathrm{~cm}\) from its relaxed length when a force of \(7.50 \mathrm{~N}\) is applied. A \(0.500-\mathrm{kg}\) particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to \(x=5.00 \mathrm{~cm}\) and released from rest at \(t=0\). (a) What is the force constant of the spring? (b) What are the angular frequency \(\omega\), the frequency, and the period of the motion? (c) What is the total energy of the system? (d) What is the amplitude of the motion? (e) What are the maximum velocity and the maximum acceleration of the particle? (f) Determine the displacement \(x\) of the particle from the equilibrium position at \(t=0.500 \mathrm{~s}\). (g) Determine the velocity and acceleration of the particle when \(t=0.500 \mathrm{~s}\).

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