16.3.2 The Mathematics of Sinusoidal Waves

FIGURE 16.13 shows a snapshot graph at $t = 0$ of a sinusoidal wave. The sinusoidal function that describes the displacement of this wave is

D (x, t = 0) = A sin (2 π \frac{x}{λ} + ϕ_{0})

(16.7)

where the notation $D (x, t = 0)$ means that we’ve frozen the time at $t = 0$ to make the displacement a function of only x. The term $ϕ_{0}$ is a phase constant that characterizes the initial conditions. (We’ll return to the phase constant momentarily.)

The figure illustrates a snapshot graph of a wave at t equals 0.

FIGURE 16.13 Full Alternative Text

The function of Equation 16.7 is periodic with period $λ$ . We can see this by writing

\begin{array}{l} D (x + λ) & = A \sin (2 π \frac{(x + λ)}{λ} + ϕ_{0}) = A \sin (2 π \frac{x}{λ} + ϕ_{0} + 2 π rad) \\ = A \sin (2 π \frac{x}{λ} + ϕ_{0}) = D (x) \end{array}

where we used the fact that $sin (a + 2 π rad) = s i n a$ . In other words, the disturbance created by the wave at $x + λ$ is exactly the same as the disturbance at x.

The next step is to set the wave in motion. We can do this by replacing x in Equation 16.7 with $x - v t$ . To see why this works, recall that the wave moves distance vt during time t. In other words, whatever displacement the wave has at position x at time t, the wave must have had that same displacement at position $x - v t$ at the earlier time $t = 0$ . Mathematically, this idea can be captured by writing

D (x, t) = D (x - v t, t = 0)

(16.8)

Make sure you understand how this statement describes a wave moving in the positive x-direction at speed v.

This is what we were looking for. $D (x, t)$ is the general function describing the traveling wave. It’s found by taking the function that describes the wave at $t = 0$ —the function of Equation 16.7—and replacing x with $x - v t$ . Thus the displacement equation of a sinusoidal wave traveling in the positive x-direction at speed v is

D (x, t) = A sin (2 π \frac{x - v t}{λ} + ϕ_{0}) = A sin (2 π (\frac{x}{λ} - \frac{t}{T}) + ϕ_{0})

(16.9)

In the last step we used $v = λ f = λ / T$ to write $v / λ = 1 / T$ . The function of Equation 16.9 is not only periodic in space with period $λ$ , it is also periodic in time with period T. That is, $D (x, t + T) = D (x, t)$ .

It will be useful to introduce two new quantities. First, recall from simple harmonic motion the angular frequency

ω = 2 π f = \frac{2 π}{T}

(16.10)

The units of $ω$ are $rad / s$ , although many textbooks use simply $s^{- 1}$ .

You can see that $ω$ is $2 π$ times the reciprocal of the period in time. This suggests that we define an analogous quantity, called the wave number k, that is $2 π$ times the reciprocal of the period in space:

k = \frac{2 π}{λ}

(16.11)

The units of k are $rad / m$ , although many textbooks use simply $m^{- 1}$ .

NOTE The wave number k is not a spring constant, even though it uses the same symbol. This is a most unfortunate use of symbols, but every major textbook and professional tradition uses the same symbol k for these two very different meanings, so we have little choice but to follow along.

We can use the fundamental relationship $v = λ f$ to find an analogous relationship between $ω$ and k:

v = λ f = \frac{2 π}{k} \frac{ω}{2 π} = \frac{ω}{k}

(16.12)

which is usually written

ω = v k

(16.13)

Equation 16.13 contains no new information. It is a variation of Equation 16.5, but one that is convenient when working with k and $ω$ .

If we use the definitions of Equations 16.10 and 16.11, Equation 16.9 for the displacement can be written

\begin{matrix} D (x, t) = A sin (k x - ω t + ϕ_{0}) \\ (sinusoidal wave traveling in the positive x -direction) \end{matrix}

(16.14)

A sinusoidal wave traveling in the negative x-direction is $A sin (k x + ω t + ϕ_{0})$ . Equation 16.14 is graphed versus x and t in FIGURE 16.14.

The figure consists of two parts that illustrate history and snapshot curves for a sinusoidal wave.

FIGURE 16.14 Full Alternative Text

You learned in 《 SECTION 15.2 that the initial conditions of an oscillator can be characterized by a phase constant. The same is true for a sinusoidal wave. At $(x, t) = (0 m, 0 s)$ Equation 16.14 becomes

D (0 m, 0 s) = A sin ϕ_{0}

(16.15)

Different values of $ϕ_{0}$ describe different initial conditions for the wave.

EXAMPLE 16.3 Analyzing a sinusoidal wave

A sinusoidal wave with an amplitude of 1.00 cm and a frequency of 100 Hz travels at $200 m / s$ in the positive x-direction. At $t = 0 s,$ the point $x = 1.00 m$ is on a crest of the wave.

Determine the values of A, v, $λ$ , k, $f$ , $ω$ , T, and $ϕ_{0}$ for this wave.
Write the equation for the wave’s displacement as it travels.
Draw a snapshot graph of the wave at $t = 0 s$ .

VISUALIZE The snapshot graph will be sinusoidal, but we must do some numerical analysis before we know how to draw it.

SOLVE

There are several numerical values associated with a sinusoidal traveling wave, but they are not all independent. From the problem statement itself we learn that

$A = 1.00 cm v = 200 m/s f = 100 Hz$

We can then find:

$\begin{matrix} λ & = & v / f = 2.00 m \\ k & = & 2 π / λ = π rad/m or 3.14 r a d / m \\ ω & = & 2 π f = 628 rad / s \\ T & = & 1 / f = 0.0100 s = 10.0 m s \end{matrix}$

The phase constant $ϕ_{0}$ is determined by the initial conditions. We know that a wave crest, with displacement $D = A$ , is passing $x_{0} = 1.00 m$ at $t_{0} = 0 s$ . Equation 16.14 at $x_{0}$ and $t_{0}$ is

$D (x_{0}, t_{0}) = A = A sin [k (1.00 m) + ϕ_{0}]$

This equation is true only if $sin [k (1.00 m) + ϕ_{0}] = 1$ , which requires

$k (1.00 m) + ϕ_{0} = \frac{π}{2} rad$

Solving for the phase constant gives

$ϕ_{0} = \frac{π}{2} rad - (π rad/m)(1.00 m) = - \frac{π}{2} rad$
With the information gleaned from part a, the wave’s displacement is

$D (x, t) = 1.00 cm \times sin [(3.14 rad/m) x - (628 rad/s) t - π / 2 rad]$

Notice that we included units with A, k, $ω$ , and $ϕ_{0}$ .
We know that $x = 1.00 m$ is a wave crest at $t = 0 s$ and that the wavelength is $λ = 2.00 m$ . Because the origin is $λ / 2$ away from the crest at $x = 1.00 m,$ we expect to find a wave trough at $x = 0$ . This is confirmed by calculating $D (0 m, 0 s) = (1.00 cm) sin (- π / 2 rad) = - 1.00 cm$ . FIGURE 16.15 is a snapshot graph that portrays this information.

FIGURE 16.15 A snapshot graph at $t = 0 s$ of the sinusoidal wave of Example 16.3.

FIGURE 16.15 Full Alternative Text

16.3.2 The Mathematics of Sinusoidal Waves

FIGURE 16.13 A sinusoidal wave.

FIGURE 16.14 Interpreting the equation of a sinusoidal traveling wave.

EXAMPLE 16.3 Analyzing a sinusoidal wave

FIGURE 16.15 A snapshot graph at $t = 0 s$ of the sinusoidal wave of Example 16.3.