The initial condition is not required because of periodicity. Since both the equation and the boundary conditions are linear, the velocity can be written as the real part of some complex function
because .
Substituting this into the partial differential equation reduces it to ordinary differential equation
with boundary conditions
The solution to the above problem is
The disturbance created by the oscillating plate travels as the transverse wave through the fluid, but it is highly damped by the exponential factor. The depth of penetration  of this wave decreases with the frequency of the oscillation, but increases with the kinematic viscosity of the fluid.
The force per unit area exerted on the plate by the fluid is
There is a phase shift between the oscillation of the plate and the force created.