第16章波動

In the previous section, we described periodic waves by their characteristics of wavelength, period, amplitude, and wave speed of the wave. Waves can also be described by the motion of the particles of the medium through which the waves move. The position of particles of the medium can be mathematically modeled as wave functions, which can be used to find the position, velocity, and acceleration of the particles of the medium of the wave at any time.

A pulse can be described as wave consisting of a single disturbance that moves through the medium with a constant amplitude. The pulse moves as a pattern that maintains its shape as it propagates with a constant wave speed. Because the wave speed is constant, the distance the pulse moves in a time $\text{Δ}t$ is equal to $\text{Δ}x=v\text{Δ}t$ (Figure 16.8).

Consider a string kept at a constant tension $F_T$ where one end is fixed and the free end is oscillated between $y=\text{+}A$ and $y=\text{−}A$ by a mechanical device at a constant frequency. Figure 16.9 shows snapshots of the wave at an interval of an eighth of a period, beginning after one period $\left(t=T\right).$

Notice that each select point on the string (marked by colored dots) oscillates up and down in simple harmonic motion, between $y=+A$ and $y=\text{−}A,$ with a period T. The wave on the string is sinusoidal and is translating in the positive x-direction as time progresses.

At this point, it is useful to recall from your study of algebra that if f(x) is some function, then $f\left(x-d\right)$ is the same function translated in the positive x-direction by a distance d. The function $f\left(x+d\right)$ is the same function translated in the negative x-direction by a distance d. We want to define a wave function that will give the y-position of each segment of the string for every position x along the string for every time t.

Looking at the first snapshot in Figure 16.9, the y-position of the string between $x=0$ and $x=\lambda$ can be modeled as a sine function. This wave propagates down the string one wavelength in one period, as seen in the last snapshot. The wave therefore moves with a constant wave speed of $v=\lambda \text{/}T.$

Recall that a sine function is a function of the angle $\theta$, oscillating between $\text{+}1$ and $\mathrm{-1}$, and repeating every $2\pi$ radians (Figure 16.10). However, the y-position of the medium, or the wave function, oscillates between $\text{+}A$ and $\text{−}A$, and repeats every wavelength $\lambda$.

To construct our model of the wave using a periodic function, consider the ratio of the angle and the position,

Using $\theta =\frac{2\pi }{\lambda }x$ and multiplying the sine function by the amplitude A, we can now model the y-position of the string as a function of the position x:

The wave on the string travels in the positive x-direction with a constant velocity v, and moves a distance vt in a time t. The wave function can now be defined by

It is often convenient to rewrite this wave function in a more compact form. Multiplying through by the ratio $\frac{2\pi }{\lambda }$ leads to the equation

The value $\frac{2\pi }{\lambda }$ is defined as the wave number. The symbol for the wave number is k and has units of inverse meters, ${\text{m}}^{\mathrm{-1}}:$

Recall from Oscillations that the angular frequency is defined as $\omega \equiv \frac{2\pi }{T}.$ The second term of the wave function becomes

The wave function for a simple harmonic wave on a string reduces to

where A is the amplitude, $k=\frac{2\pi }{\lambda }$ is the wave number, $\omega =\frac{2\pi }{T}$ is the angular frequency, the minus sign is for waves moving in the positive x-direction, and the plus sign is for waves moving in the negative x-direction. The velocity of the wave is equal to

Think back to our discussion of a mass on a spring, when the position of the mass was modeled as $x\left(t\right)=A\text{cos}\left(\omega t+\varphi \right).$ The angle $\varphi$ is a phase shift, added to allow for the fact that the mass may have initial conditions other than $x=\text{+}A$ and $v=0.$ For similar reasons, the initial phase is added to the wave function. The wave function modeling a sinusoidal wave, allowing for an initial phase shift $\varphi ,$ is

The value

is known as the phase of the wave, where $\varphi$ is the initial phase of the wave function. Whether the temporal term $\omega t$ is negative or positive depends on the direction of the wave. First consider the minus sign for a wave with an initial phase equal to zero $\left(\varphi =0\right).$ The phase of the wave would be $\left(kx-\omega t\right).$ Consider following a point on a wave, such as a crest. A crest will occur when $\text{sin}\left(kx-\omega t\right)=1.00$, that is, when $kx-\omega t=n\pi +\frac{\pi }{2},$ for any integral value of n. For instance, one particular crest occurs at $kx-\omega t=\frac{\pi }{2}.$ As the wave moves, time increases and x must also increase to keep the phase equal to $\frac{\pi }{2}.$ Therefore, the minus sign is for a wave moving in the positive x-direction. Using the plus sign, $kx+\omega t=\frac{\pi }{2}.$ As time increases, x must decrease to keep the phase equal to $\frac{\pi }{2}.$ The plus sign is used for waves moving in the negative x-direction. In summary, $y\left(x,t\right)=A\text{sin}\left(kx-\omega t+\varphi \right)$ models a wave moving in the positive x-direction and $y\left(x,t\right)=A\text{sin}\left(kx+\omega t+\varphi \right)$ models a wave moving in the negative x-direction.

Equation 16.4 is known as a simple harmonic wave function. A wave function is any function such that $f\left(x,t\right)=f\left(x-vt\right).$ Later in this chapter, we will see that it is a solution to the linear wave equation. Note that $y\left(x,t\right)=A\text{cos}\left(kx+\omega t+\varphi \text{′}\right)$ works equally well because it corresponds to a different phase shift $\varphi \text{′}=\varphi -\frac{\pi }{2}.$