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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 $x\left(t\right)=A\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\omega t+\varphi \right).$ 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 $\text{sin}\phantom{\rule{0.2em}{0ex}}\left(kx-\omega t\right)=1.00$ , 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, $y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t+\varphi \right)$ models a wave moving in the positive x -direction and $y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx+\omega t+\varphi \right)$ models a wave moving in the negative x -direction.
[link] 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 $y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{cos}\left(kx+\omega t+\varphi \text{\u2032}\right)$ works equally well because it corresponds to a different phase shift $\varphi \text{\u2032}=\varphi -\frac{\pi}{2}.$
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