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Chapter Review

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In this chapter we study waves. You can view a wave as a result from the connection of a series of oscillators (oscillations were studied in the previous chapter) or as a propagating oscillation. Most generally, any disturbance that propagates can be called a wave. In this chapter we focus on harmonic waves in which the oscillation that gives rise to the wave is a simple harmonic oscillation. The study of waves is important in almost every branch of physics and has many applications.

14-1 Types of Waves

There are two main types of waves. These types are distinguished by the relationship between the direction of the oscillation of the medium in which the wave travels and the direction of the propagation of the wave. In a transverse wave the direction of oscillation is perpendicular (or transverse) to the direction of propagation. A wave on a string is a good example of a transverse wave. The other main type of wave is a longitudinal wave in which the direction of oscillation is along the same line as the direction of propagation. A compression wave traveling along a spring (such as a slinky) is a good example of a longitudinal wave. Since a harmonic wave results from the simple harmonic motion within a medium, the main characteristics of waves are related to the cycle of this repeating motion. One of these characteristics relates to the minimum time it takes for a wave to repeat itself, the period T. As with any simple harmonic motion the inverse of the period is called the frequency f.

Waves also repeat themselves spatially; the minimum repeat length of a wave is called its wavelength l. If you consider the picture of a transverse wave below, the wavelength equals the distance between successive crests, or troughs, of the wave (other corresponding points may also be used).

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A third important characteristic of a wave is its speed of travel. This speed equals the distance the wave travels before it repeats (the wavelength) divided by the time it takes the wave to repeat (the period). Therefore, the speed of a wave is given by

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Practice Quiz

The distance between two troughs of a wave of frequency 200 Hz is 1.7 m, what is the speed of this wave? 5.0 mm/s 340 m/s 8.5 mm/s 120 m/s [None of the above]

If two waves traveling through the same medium have the same period, but wave 1 has half the wavelength of wave 2, how do the speeds of these waves compare? v1 = 2v2 v1 = 4v2 v2 = 4v1 v2 = 2v1 v2 = v1

14-2 Waves on a String

Waves traveling on a string (or any sort of linear cord) is a common way that sounds are generated and so it is important to understand this behavior. The speed at which a wave on a string travels is determined by two properties of the string: its tension and linear m (mass per unit length). The more tightly pulled the string is, the more rapidly it will oscilate and the faster the wave will travel. The heavier a segment of the string is, the more sluggishly (or slowly) it will move under a given tension and the slower the wave will travel. Detailed analysis of this situation shows that the relationship between these three quantities is

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where F is the tension in the string (force transmitted through it) and the mass per unit length is given by m = m/L with m and L being the mass and length of the string, respectively.

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Physlet Illustration: The Speed of a Wave on a String
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Tension = Newtons Mass/Length =  g/m
A wave pulse moves on a string as shown above.  The grid spacing is in meters and the times are shown in seconds. Vary the tension in the string (from 0 to 100 N) and the mass per unit length of the string (from 10 to 100 g/m) and determine the relationship between their values and the wave speed.  Is it correct?  Start
Hints Measure the position of the pulse at several times and determine the wave speed.   Does the wave speed increase or decrease when the tension is increased? Does the wave speed increase or decrease when the mass per unit length is increased? Is the wave speed equal to the tension divided by the mass per unit length?

Example 14.1 Sending a Wave: A string of length 2.59 m has a mass of 5.11 and is fixed at one end. A person takes the other end and oscillates it up and down with a frequency of 3.47 Hz. If it takes the resulting wave 0.862 s to travel the length of the string, (a) determine the tension in the string, and (b) determine the wavelength of the wave.

Picture the Problem The picture shows the person oscillating one end of the string while the other end remains fixed.

Strategy The given information allows us to determine both v and m so that we can use the relationship between v, m, and F to solve part (a). Knowing v and frequency would then allow us to calculate the wavelength.

Solution

Part (a)

1. Determine mass per unit length:
2. Determine the speed of the wave:
3. Solve for tension F:
4. Obtain the numerical result:
Part (b)
1. Use the relationship between v and l to solve for l:

Insight Notice how the wave speed connects the properties of waves (f, l) to the properties of the medium (F, m).

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In the above example a wave was sent along a string with a fixed end. When the wave reaches that end it will be reflected back in the opposite direction along the string. Because the end is fixed it inverts each wave pulse upon reflection as a result of applying a force in a direction opposite to the force applied by the string on the fixed connection. If the end of the string opposite the person was free to move the reflected wave would not be inverted relative to the initial wave because the end oscillates along with the rest of the string. This issue of how a wave is reflected will become important in later discussions.

Practice Quiz

A certain string sustains waves of speed v when under a tension F. If the tension in the string is reduced to , is the speed of the wave reduced or enhanced and by what factor is it reduced or enhanced? reduced by a factor of 0.58 reduced by a factor of 3 enhanced by a factor of 1.7 enhanced by a factor of 3 the speed stays the same

14-3* Harmonic Wave Functions

A traveling harmonic wave can be described by a reasonably simple functional form. For clarity we will consider a transverse wave for which the direction of oscillation is the y-direction and the direction of propagation is the x-direction. The position of a point on the wave (i.e., y) depends both on where you look in space (i.e., on x) and when you look in time (i.e., on t). So, the position y is a function of both x and t. A harmonic wave traveling in the +x-direction can be described by the equation

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In this equation, A is the amplitude of the wave (it's maximum displacement from equilibrium). Examination of this expression shows that, for fixed t (a snapshot of the wave), the wave repeats whenever x increases by an amount l. Similarly, for fixed x (at a given location) the wave repeats whenever t increases by an amount T.

Physlet Illustration: Basic Wave Properties
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A wave travels through a medium. The grid display is in meters, and the times are shown in seconds. What are the wave's wavelength, frequency and speed? Start
Hints: Stop the animation at a convenient time, and measure the peak-to-peak separation on the horizontal grid. This distance is the wavelength, l, in meters. Let the animation run and choose a single point along the wave's path (e.g. the point x = 0). Try to determine how long it takes between one crest and the next striking that point. This time is called the period, T, in seconds.  The inverse of the period, 1/T,  is the frequency, f, in Hz. Finally, choose one point on the wave (e.g. one particular wave crest) to follow, and measure how far that point moves in a particular time interval. In this way, you determine the wave's speed, v, in m/s. According to the theory of traveling waves, how should these three quantities (l, f and v) be related mathematically? Do your results confirm this relationship?

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Example 14.2 A Harmonic Wave: A transverse harmonic wave is described by the function

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What is the frequency of this wave?

Solution:

By comparing the given function to the general form we can determine that . This means that

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Since frequency is given by f= 1/T we can conclude

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Practice Quiz

What is the speed of a wave described by the expression ? 48 m/s 0.38 m/s 18 m/s 0.21 m/s 2.4 m/s

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