Traveling Wave Formula

Thus far we have formulas for a time varying sine wave sin ωt , and a space varying wave sin kx. Now we want a formula for a traveling sine wave whose amplitude varies in both space and time. The answer turns out to be
..... (1)
What we will do is show that this formula represents a sine wave moving down the x axis. Figure (11) shows a sinusoidal shape that is moving down the x axis at a speed v_wave . If we describe the wave by the function sin θ , then it is the origin sin θ = 0 that moves down the axis at a speed v_wave . Thus what we need is a formula for θ so that when we set θ = 0 , that point does move down the x axis at the desired speed. The answer we gave in Equation 1 suggests that the correct formula for θ is

θ = kx – ωt (2)
Setting θ = 0 we get
 .....(3)
But if the θ = 0 point travels at a speed v_wave , then after a time t, it has traveled a distance x given by
x = v_wavet (4)

Comparing Equations (2) and (3), we see that the point θ = 0 moves along the x axis at a speed
(4)
You recognize that the quantity
ω/ k has the dimensions of a velocity
...... (5)
Thus the origin does move down the x axis at a speed v_wave , and the formula (kx – ωt) , is our desired traveling wave formula.

Figure 1: The cycle begins at θ = 0.