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Speed of Travelling wave

Figure shows two snapshots of the wave of equation taken a small time interval Δ t apart.

ure-Two snapshots of a wave
taken a small time interval Δ t apart.

y (x, t) = ym sin (kx – ω t)                -------(i)

The wave is traveling in the direction of increasing x (to the right in figure), the entire wave pattern moving a distance Δ x in that direction during the interval Δ t. The ratio Δ x/Δ t (or, in the differential limit, dx / dt) is the wave speed v. How can we find its value?

As the wave in figure moves, each point of the moving wave form (such as point A) retains its displacement y. (Points on the string do not retain their displacement, but points on the wave form do). For each such points Equation (i) tells us that the argument of the sine function must be a constant.


kx - ω t = a constant                         -------- (ii)

Note that although this argument is constant, both x and t are changing. In fact, as t increase, x must also, to keep the argument constant. This confirms that the wave pattern is moving toward increasing x.
To find the wave speed v, as take the derivative of Equation (ii), getting

K = 0

ω =


Using the equations k=2π /λ and ω = 2π /), we can rewrite the wave speed as
v =

The equation v = λ /T tells us that the wave speed is one wavelength per period: the wave moves a distance of one wavelength in one period of oscillation.

Equation (i) describes a wave moving in the direction of increasing x. We can find the equation of a wave traveling in the opposite direction by replacing t in Equation (i) with –t. This corresponds to the condition
kx + ω t = a constant

Which (compare Equation (ii)) requires that x decrease with time. Thus, a wave traveling toward decreasing x is described by the equation

y (x, t) = ym sin (kx + ω t)              ---------(iii)

If you analyze the wave of equation (iii) as we have just done for the wave of Equation (i), you will find for its velocity

The minus sign verifies that the wave is indeed moving in the direction of decreasing x and justifies our switching the sign of the time variable.
Constant now a wave of generalized shape, given by
Y (x, t) =
η (kx ± ω t)                     -------- (iv)

η represents any function, the sine function being one possibility. Our analysis above shows that all waves in which the variables x and t enter in the combination kx ± ω t are traveling waves. Furthermore, all traveling waves must be of the form of Equation (iv).

Thus y (x, t) = represents a possible (though perhaps physically a little bizarre) traveling wave. The function y (x, t) = sin(ax2 – bt), on the other hand, does not represent a traveling wave.

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