A continuous succession of sinusoidal wave pulses are...

A continuous succession of sinusoidal wave pulses are produced at one end of a very long string and travel along the length of the string. The wave has frequency 70.0 Hz, amplitude 5.00 mm, and wavelength 0.600 m. (a) How long does it take the wave to travel a distance of 8.00 m along the length of the string? (b) How long does it take a point on the string to travel a distance of 8.00 m, once the wave train has reached the point and set it into motion? (c) In parts (a) and (b), how does the time change if the amplitude is doubled?

Key Concepts

Simple Harmonic Motion

Simple harmonic motion describes the oscillatory behavior of individual particles in a medium when a sinusoidal wave passes through. While the wave pulse travels along the string at a constant speed, the particles themselves undergo periodic oscillations about their equilibrium positions. These oscillations have specific amplitudes and periods, which are determined by the wave's properties, and the motion of an individual point on the string is characterized by this back-and-forth movement rather than a unidirectional displacement.

Amplitude and Its Effects

Amplitude is the maximum displacement of a particle from its equilibrium position in the context of wave motion. While amplitude affects the energy carried by the wave and the extent of oscillation of individual particles, it does not influence the wave’s propagation speed or the oscillation period in linear, ideal systems. This concept is important when considering problems where changes in amplitude are examined relative to the time or distance characteristics of both wave propagation and particle motion.

Wave Propagation

Wave propagation refers to the movement of energy through a medium via oscillations or disturbances. In the context of a sinusoidal wave on a string, the wave travels at a speed determined by the relationship between its frequency and wavelength. This speed tells us how quickly the wave reaches a certain point along the string, independent of the oscillatory motion of the particles in the medium.

Frequency and Wavelength Relationship

The relationship between frequency and wavelength is fundamental to understanding the speed of a wave. The wave speed is given by the product of these two quantities. An increase in either frequency or wavelength, while the other remains constant, directly affects the distance a wave pulse covers in a given time, which is critical when calculating the time taken for a pulse to travel a specific distance along a medium.

Transcript

00:01 So the first part is asking us how to find, whether the amount of time it takes for the wave to travel eight meters.

00:07 And then the second part of the problem is asking us how long would it take for a point on the wave to travel eight meters? so they're very distinct questions.

00:16 Let's start with part a.

00:17 Part a, we can simply find the speed of propagation of this wave by multiplying the wavelength and the frequency.

00:23 The wavelength is going to be 0 .6 meters.

00:26 And then the frequency is given of 70 .0 hertz.

00:33 And this is going to give us a speed of propagation of 42 meters per second.

00:40 Time taken is simply going to be the distance over the speed.

00:49 And they want us to find a distance of eight.

00:51 So we'll have eight meters.

00:53 And then this will be divided by 42 meters per second.

00:58 And this is going to be 0 .19 seconds.

01:02 So this will be the answer for the first part of the question.

01:06 It takes 0 .19 seconds for this wave to travel 8 meters.