Question

Show that the equation,y=a sin(omegat-kx) satisfies the wave equation(del^(2) y)/(delt^(2))=v^(2) (del^(2)y)/(delx^(2)). Find speed of wave and the direction in which it is travelling.

Answer 1

We are given equation of the wave as ,

• y = asin(ωt - kx )

(1) Differentiating the above equation w.r.t. t,

• $\frac{dy}{dt}$ = acos(ωt - kx ).ω

(2) Differentiating is again w.r.t. t,

• $\frac{d^{2}y }{dt^{2} }$ = - asin(ωt - kx )ω²    ....(a)

(3) Now, differentiating the equation of the wave w.r.t. x ,

• $\frac{dy}{dx}$ = acos(ωt - kx)(-k)

(4) Differentiating again w.r.t. x,

• $\frac{d^{2}y }{dx^{2} }$ = -asin(ωt - kx)(-k)(-k)

             = -asin(ωt - kx)(k²)     ....(b)

(5) The given wave equation is ,

• $\frac{d^{2}y }{dt^{2} } = v^{2} \frac{d^{2}y }{dx^{2} }$

(6) Comparing the above equation with equations (a) and (b),

• We get , $v^{2} = \frac{w^{2} }{k}$  

•               $v = \frac{w}{k}$    this is the speed of the wave
• In the equation of the wave, there is a negative sign between ωt and kx. This implies the wave is travelling in the positive x direction.