@Carl Love  Many thanks again.  For the first plot, this is exactly the plot I wanted to get to, however, I see that it is only plotted as a function of z, not z/b.  Is this possible?

For the animation, since I cannot obtain values for K[n], I was thinking I could plot y_n(z,t)/K[n] as a function of t.  Is this possible? Sorry, I should have been more clear, it was an animation as a function of t.

Thanks again, I really appreciate your help.

@Preben Alsholm This is exactly what I tried, but it didn't work.  Perhaps because I am a new user?

Apologies if the attached file is not shown - I can't seem to upload it.  The content is illustrated below:

(1)The function to animate is $$y_n(z,t) = K_n \cos(a_n^2 \sqrt{g} t - \sigma_n) J_o(c_n \sqrt{z/b})$$ for $n$ from 1 to 4 where $J_o$ is the bessel function of the first kind and $c_n$ are the roots of the Bessel function, $a_n^2 = c_n^2/4b$, $\sqrt{g}$ a constant and $\delta_n$ a shift,

(2)The function I want to plot is $$Z(z) = J_o (c_n \sqrt{z/b})$$ as a function of $z/b$ for $0<z<b$ if $b=100$

@Carl Love  Many thanks, I will implement this today.  The equation obtained can be further reduced to the Bessel equation of zero order.  I was wondering if I could ask your interpretation on something;  Given the solution X(z,t) = \sum_{n=1}^{\infty} J_o (c \sqrt{z/b}) K_n \cos(a_n d t - \sigma_n), which is a function with a shape modelled by the Bessel function (J_o) of the first kind, zero order and a time-varying amplitude part (K_n...) where d is a constant and \sigma_n is a shift, I want to plot the shape as function of z/a for 2 values of n.

My question is: Should I just plot the bessel part of the function? I would have plotted the whole function, but I am not provided with any initial conditions except that y(z,0)=F(z) and y'(z,0)=G(z) and F and G are not provided.  Since these are not provided, in the equation to determine the factor K_n, I have nothing to substitude for F and G and so cannot compute the coefficients.

K_n = \sqrt{p_n + q_n} where p_n and q_n are the coefficients of cos and sin when I expand cos(a_n dt - \sigma_n).

Thanks for any help.