Sinusoidal wave equation

We found that the disturbance (whether pulse or wave, transverse or longitudinal) depends on both position x and time t. If we call the displacement y, we can write y = f(x,t) or y(x,t) to represent this functional dependence on time and position. In the example of the transverse pulse traveling along a Slinky as pictured in Fig. 17-2,

y(x,t) represents the transverse (vertical) displacement of the Slinky rings from their equilibrium position at given position x and time t. (Alternatively, in the longitudinal wave on the Slinky shown in Fig. 17-1, y(x,t) could represent the number of Slinky coils per centimeter at a given x and t.)
We can completely describe any wave or pulse that does not change shape over time and travels at a constant velocity using the relation y = f(x,t), in which y is the displacement as a function/of the time t and the position x. In general, a wave can have any shape so long as it is not too sharp. The trick then is to find the correct expression for the function, f(x,t).

Fortunately, it turns out that any shape pulse or wave can be constructed by adding up different sinusoidal oscillations. This makes the description of sinusoidal waves especially useful. So, for the rest of this section we'll discuss the properties and descriptions of continuous waves produced by displacing a stretched string using a sinusoidal motion like that shown in Fig. 17-3b. We will start by using the equation we developed in Chapter 16 to describe for sinusoidal motion at the location of a single piece of string. As we did in looking through the slit in Fig. 17-5, we will only let time vary. Next we can consider how to describe a snapshot that records the displacement of many pieces of the string at a single time. Finally, we can combine our snapshot with the results of peeking through a slit to get a single equation that ought to describe the propagation of a single sinusoidal wave. Basically we are trying to describe the displacement y of every piece of the string from its equilibrium point at every time. We are looking for y(x,t).

Looking Through a Slit: Sinusoidal Wave Displacement at x = 0
If we choose a coordinate system so that x = 0 m at the left end of the string in Fig. 17-3b, then the motion at the left end of the string can be described using Eq. 16-5 with the string displacement from equilibrium represented by y(x,t) = y(0,r) rather than by simply y(t). To simplify our consideration we assume that the initial phase of the string oscillation at x = 0 m and t = 0 s is zero. This gives us

where the angular frequency can be related to the period of oscillation by to = 2irlT. Although we use the cosine function in Chapter 16 to describe simple harmonic motion, it is customary to use the sine function to describe wave motion. As we mentioned in Chapter 16, when a sine function is shifted to the left by v/2 it looks like a cosine function. So we can also describe the same string displacement as a function of time at x = 0 m as

Note that using the sine function requires a different, nonzero initial phase angle given by ir!2. If we locate our slit at another nonzero value of x as shown in Fig. 17-5, then the initial phase (at t = 0 s) will often turn out to be different from w/2. In fact this initial phase is a function of the location x of the piece of string we are considering.
A Snapshot: Sinusoidal Wave Displacement at t = 0

Imagine that the man has been moving the end of the string up and down as shown in Fig. 17-36 for a long time using a sinusoidal motion. Instead of looking through a slit as time varies, we take a snapshot of the string at a time t = Os similar to that shown in Fig. 17-36. Then we expect our snapshot to be described by the equation

where A: is a constant and the "initial" phase when x is zero must also be tt!2. Note that if the snapshot of the string were taken at another time, the initial phase would probably be different.

Combining Expressions for x and t
Equation 17-1 describes the displacement at all times for just the piece of string located at x = 0 m. Equation 17-2 describes the displacement of all the pieces of string at t = Os. We can make an intelligent guess that the equation describing y(x,t) is some combination of these two expressions given by
y(x,t) = Ysin[(foc ± cot) + tt/2)], (17-3)

where tt/2 represents the initial phase when x = 0 m and t - 0 s for the special case we considered. 

In general we can describe the motion of our sinusoidal wave with an arbitrary initial phase by modifying Eq. 17-3 to get
 y(x,t) = Ysin[(fct ± tot) + 4>q)] (sinusoidal wave motion, arbitrary initial phase), (17-4)
where <f>₀ is the initial phase (or phase constant) when both x = Om and t = Os. The ± sign refers to the direction of motion of the wave as we shall see in Section 17-5. In cases where the initial phase is not important, we can simplify Eq. 17-3 by choosing an initial time and origin of the x axis that lies along the line of motion of the wave so that <t>o = 0 rad.