Question

3. By separating the variables, show that the one-dimensional wave equation \frac{\partial^{2}z}{\partial x^{2}}=\frac{1}{c^{2}}\frac{\partial^{2}z}{\partial t^{2}} has solution of the form Ae^{(\pm imttmit)}

Answer 1

Answer:

Consider the relation between Newton’s law that is applied to the volume ΔV in the direction x:

ΔF=Δmdvxdt (Newton’s law)

Where,

F: force acting on the element with volume ΔV

ΔFx=−ΔpxΔSx =(∂p∂xΔx+∂p∂xdt)ΔSx ≃−∂p∂xΔV −ΔV∂p∂x=Δmdvxdt (as dt is small, it is not considered and ΔSx is in x direction so ΔyΔz and from Newton’s law)

=ρΔVdvxdt

From dvxdtas∂vx∂t dvxdt=∂vx∂t+vx∂vx∂x≈∂vx∂x −∂p∂x=ρ∂vx∂t

Above equation is known as the equation of motion.

−∂∂x(∂p∂x)=∂∂x(ρ∂vx∂t) =ρ∂∂t(∂vx∂x) −∂2p∂x2=ρ∂∂t(−1K∂p∂t) (from conservation of mass)

∂p2∂x2−ρK∂2p∂t2=0

Where,

K: bulk modulus

Rewriting the above equation:

∂p2∂x2−1c2∂2p∂t2=0

Where,

c: velocity of sound given as c=Kρ−−√

Thus, above is the one-dimensional wave equation derivation.