chapter 16 wave motion

16.1 Propagation of a Disturbance

In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.

Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass transport. They consist, instead, of oscillations or vibrations around almost fixed locations. For example, a cork on rippling water will bob up and down, staying in about the same place while the wave itself moves onwards.

One type of wave is a mechanical wave, which propagates through a medium in which the substance of this medium is deformed. The deformation reverses itself owing to restoring forces resulting from its deformation. For example, sound waves propagate via air molecules bumping into their neighbors. This transfers some energy to these neighbors, which will cause a cascade of collisions between neighbouring molecules. When air molecules collide with their neighbors, they also bounce away from them (restoring force). This keeps the molecules from continuing to travel in the direction of the wave.

Another type of wave can travel through a vacuum, e.g. electromagnetic radiation (including visible light, ultraviolet radiation, infrared radiation, gamma rays, X-rays, and radio waves). This type of wave consists of periodic oscillations in electrical and magnetic fields.

A main distinction can be made between transverse and longitudinal waves. Transverse waves occur when a disturbance sends waves perpendicular (at right angles) to the original wave. Longitudinal waves occur when a disturbance sends waves in the same direction as the original wave.

Waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this equation varies depending on the type of wave.

Two types of waves :

•      Mechanical waves - some physical medium is being disturbed

•      Electromagnetic waves - do not require a medium to propagate

* the essence of wave motion—the transfer of energy through space without the accompanying transfer of matter

All mechanical waves require :

•      (1) some source of disturbance,

•      (2) a medium that can be disturbed, and

•      (3) some physical mechanism through which elements of the medium can influence each other.

One way to demonstrate wave motion is to flick one end of a long rope that is under tension and has its opposite end fixed  *a single bump (called a pulse) is formed and travels along the rope with a definite speed.

If we were to move the end of the rope up and down repeatedly, we would create a traveling wave, which has characteristics that a pulse does not have.

A traveling wave or pulse that causes the elements of the medium to move parallel to the direction of propagation is called a longitudinal wave.

A traveling wave or pulse that causes the elements of the disturbed medium to moveperpendicular to the direction of propagation is called a transverse wave.

The left end of the spring is pushed briefly to the right and then pulled briefly to the left. This movement creates a sudden compression of a region of the coils.

Consequently, an element of the string at x at this time has the same y position as an element located at x -vthad at time t =0:

In general, then, we can represent the transverse position y for all positions and times,measured in a stationary frame with the origin at O, as

Similarly, if the pulse travels to the left, the transverse positions of elements of the string are described by

The function y, sometimes called the wave function, depends on the two variablesx and t. For this reason, it is often written y(x, t), which is read “y as a function of xand t.”

It is important to understand the meaning of y. Consider an element of the stringat point P, identified by a particular value of its x coordinate. As the pulse passesthrough P, the y coordinate of this element increases, reaches a maximum, and thendecreases to zero. The wave function y(x, t) represents the y coordinate—thetransverse position—of any element located at position x at any time t. Furthermore,ift is fixed (as, for example, in the case of taking a snapshot of the pulse), thenthe wave function y(x), sometimes called the waveform, defines a curve representingthe actual geometric shape of the pulse at that time.

16.2 Sinusoidal Waves

On a rope, a sinusoidal wave could be established by shaking the end of the rope up and down in simple harmonic motion.The same information is more often given by the inverse of the period, which iscalled the frequency f. In general, the frequency of a periodic wave is the numberf crests (or troughs, or any other point on the wave) that pass a given point in aunit time interval. The frequency of a sinusoidal wave is related to the period by theexpression

where the constant A represents the wave amplitude and the constant  is the wavelength.We see that the vertical position of an element of the medium is the samewhenever x is increased by an integral multiple of l. If the wave moves to the right witha speed v, then the wave function at some later time t is

That is, the traveling sinusoidal wave moves to the right a distance vtin the time t, Note that the wave function has the form f (x -vt) .Ifthe wave were traveling to the left, the quantity x -vtwould be replaced by x +vt,

By definition, the wave travels a distance of one wavelength in one period T. Therefore,the wave speed, wavelength, and period are related by the expression

Substituting this expression for v       

Angular wave number

Angular frequency

wave function for sinusoidal wave

speed of sinusoidal wave

16.3 The Speed of Waves on Strings

According to Newton’s second law, the acceleration of the element increases with increasing tension. If the element returns to equilibrium more rapidly due to this increased acceleration, we would intuitively argue that the wave speed is greater. Thus, we expect the wave speed to increase with increasing tension.

Likewise, the wave speed should decrease as the mass per unit length of the string increases. This is because it is more difficult to accelerate a massive element of the string than a light element. If the tension in the string is T and its mass per unit length is μ (Greek mu), then as we shall show, the wave speed is

(16.18)

First, let us verify that this expression is dimensionally correct. The dimensions of T are ML/T² and the dimension of μ are M/L.Therefore, the dimensions of T/μ are L²/T². hence, the dimensions of  are L/T, the dimension of speed.

Consider a pulse moving on a taut string to the right with a uniform speed v measured relative to a stationary frame of reference. Instead of staying in this reference frame, it is more convenient to choose as our reference frame one that moves along with the pulse with the same speed as the pulse, so that the pulse is at rest within the frame. This change of reference frame is permitted because Newton’s laws are valid in either a stationary frame or one that moves with constant velocity. In our new reference frame, all elements of the string move to the left—a given element of the string initially to the right of the pulse moves to the left, rises up and follows the shape of the pulse, and then continues to move to the left. Figure 16.11a shows such an element at the instant it is located at the top of the pulse.

The small element of the string of length ∆s shown in Figure 16.11a, and magnified in Figure 16.11b, forms an approximate arc of a circle of radius R. In our moving frame of reference (which is moving to the right at a speed v along with the pulse), the shaded element is moving to the left with a speed v. This element has a centripetal acceleration equal to v²/R, which is supplied by components of the force T whose magnitude is the tension in the string. The force T acts on both sides of the element and is tangent to the arc, as shown in Figure 16.11b. The horizontal components of T cancel, and each vertical component T sin θ acts radially toward the center of the arc.

Hence, the total radial force on the element is 2T sin θ. Because the element is small, θ is small, and we can use the small-angle approximation sin θ ≈ θ. Therefore, the total radial force is

The element has a mass m =μ.s. Because the element forms part of a circle and subtends an angle 2θ at the center, ∆s = R(2θ), we find that

If we apply Newton’s second law to this element in the radial direction, we have.

This expression for v is Equation 16.18. Notice that this derivation is based on the assumption that the pulse height is small relative to the length of the string. Using this assumption, we were able to use the approximation sin θ ≈ θ. Furthermore, the model assumes that the tension T is not affected by the presence of the pulse; thus, T is the same at all points on the string. Finally, this proof does not assume any particular shape for the pulse. Therefore, we conclude that a pulse of any shape travels along the string with speed without any change in pulse shape 


 

16.4 Reflection and Tranmsission

          We have discussed waves traveling through a uniform medium. We now consider how a traveling wave is affected when it encounters a change in the medium. For example, consider a pulse traveling on a string that is rigidly attached to a support at one end as in Figure 16.14. When the pulse reaches the support, a severe change in the medium occurs—the string ends. The result of this change is that the pulse undergoes reflection— that is, the pulse moves back along the string in the opposite direction.

Now consider another case : this time, the pulse arrives at the end of a string that is free to move vertically, as in Figure 16.15. The tension at the free end is maintained because the string is tied to a ring of negligible mass that is free to slide vertically on a smooth post without friction. Again, the pulse is reflected, but this time it is not inverted. When it reaches the post, the pulse exerts a force on the free end of the string, causing the ring to accelerate upward. The ring rises as high as the incoming pulse,

Finally, we may have a situation in which the boundary is intermediate between these two extremes. In this case, part of the energy in the incident pulse is reflected and part undergoes transmission—that is, some of the energy passes through the boundary. For instance, suppose a light string is attached to a heavier string, as in Figure 16.16. When a pulse traveling on the light string reaches the boundary between the two, part of the pulse is reflected and inverted and part is transmitted to the heavier string. The reflected pulse is inverted for the same reasons described earlier in the case of the string rigidly attached to a support.

According to Equation 16.18, the speed of a wave on a string increases as the mass per unit length of the string decreases. In other words, a wave travels more slowly on a heavy string than on a light string if both are under the same tension. The following general rules apply to reflected waves : when a wave or pulse travels from medium A to medium B and V_A > V_B (that is, when B is denser than A), it is inverted upon reflection. When a wave or pulse travels from medium A to medium B and V_A < V_B (that is, when A is denser than B), it is not inverted upon reflection.

16.5 Rate of Energy Transfer By Sinusoidal Waves on Strings

Waves transport energy when they propagate through a medium. We can easily demonstrate this by hanging an object on a stretched string and then sending a pulse downthe string, as in Figure 16.18a. When the pulse meets the suspended object, the objectis momentarily displaced upward, as in Figure 16.18b. In the process, energy is transferredto the object and appears as an increase in the gravitational potential energy ofthe object–Earth system. This section examines the rate at which energy is transportedalong a string. We shall assume a one-dimensional sinusoidal wave in the calculation ofthe energy transferred

This expression shows that the rate of energy transfer by a sinusoidal wave on a string is proportional to (a) the square of the frequency, (b) the square of the amplitude and (c)the wave speed.

The rate of energy transfer in any sinusoidal wave is proportional to the square of  the angular frequency and to the square of the amplitude

Answer: (d). Doubling the amplitude of the wave causes the power to be larger by a factor of 4. In (a), halving the linear mass density of the string causes the power to change by a factor of 0.71—the rate decreases. In (b), doubling the wavelength of the wave halves the frequency and causes the power to change by a factor of 0.25—the rate decreases. In (c), doubling the tension in the string changes the wave speed and causes the power to change by a factor of 1.4—not as large as in part (d).

16.6 The Linier Wave Equation

All wave functions y(x, t) represent solutions of an equation calledthe linear wave equation. This equation gives a complete description of the wave motion,and from it one can derive an expression for the wave speed.Furthermore, the linearwave equation is basic to many forms of wave motion. In this section, we derive this equation as applied to waves on strings.

Suppose a traveling wave is propagating along a string that is under a tension T. Let us consider one small string element of length ∆x (Fig. 16.20). The ends of the element

make small angles θ_A and θ_B with the x axis.

Wave functions are solutions to a differential equation called the linier wave equation