Velocity of waves varying in distance and time

The amplitude y(x,t) of a sinusoidal wave varies with both distance x and time t travelled as shown in the relationship below where the wave is travelling in the forward direction.

Equation 1 y(x,t) = Ycos ( 2π t – 2π x + Φ )
T λ

Equation 2: v = λ
T

For a travelling wave, it can be shown that the distance propagated per unit time (the velocity) can be calculated from the wavelength and period. 

As shown above, the relationship (Equation 2) is the same as for other waves..
three different times

Derivation of equation 2 from equation 1:  
The distance propagated during time t is the distance between two points in the same part of the wave. y(x,t) is the same for two points on the same part of the wave:
Point 1 has coordinates x0, t0 Point 2 has the coordinates x1, t1 		  y(x1,t1) = y(x0,t0)
Substitute the two points into relationship 1 above AND
use the fact that If a = b, cos a = cos b to obtain the relationship shown at the right.
 
( 2π t1 – 2π x1 + Φ ) = ( 2π t0 – 2π x0 + Φ )
T λ T λ
For a wave of constant phase, Φ cancels from the two sides of the equation.
Dividing both sides of the equation that results by 2π gives the relationship shown at the right.
 
1 t1 – 1 x1 = 1 t0 – 1 x0
T λ T λ
Goup similar terms.
1 (t1 – t0) = 1 (x1 –x0)
T λ
Rearranging gives Equation 2.
v = λ = (x 1– x0)
T (t1 – t0)
Dr Sheila Woodgate
© The University of Auckland 2002