How to solve the dispersion relation in a symbolic expression?

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László Arany 2014-4-14

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https://ww2.mathworks.cn/matlabcentral/answers/125682-how-to-solve-the-dispersion-relation-in-a-symbolic-expression

评论： Mischa Kim 2014-4-14

采纳的回答： Mischa Kim

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I have a symbolic expression along the lines:

H = @(f) a*k*f;

where a is a constant. f is the rotational frequency and k is the wave number, which are connected through the dispersion relation:

f^2 = g*k*tanh(k*S)

where g = 9.81 is the gravitational constant and S = 20 is the water depth. I would like to plot H(f), how can I evaluate the non-linear expression k=k(f) within the symbolic expression?

Many thanks.

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Answer 1

Mischa Kim 2014-4-14

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https://ww2.mathworks.cn/matlabcentral/answers/125682-how-to-solve-the-dispersion-relation-in-a-symbolic-expression#answer_133300

编辑：Mischa Kim 2014-4-14

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Hello László, what about

 g = 9.81; S = 20; a = 1;
 H = @(k) a*k.*sqrt(g*k.*tanh(k*S));
 k = -2:0.1:2;
 plot(k,H(k))

Of course, you can first define H and f separately, if you need/want to.

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László Arany 2014-4-14

编辑：László Arany 2014-4-14

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That is a great workaround again, many thanks. In this actual case I can change the variable to k indeed.

But is it impossible to solve such an expression within a symbolic function? I mean if the expression was something like

H = @(f) f*k;

with the following relationship between the two variables:

f * tanh(f) = k * tanh(k)

then you wouldn't be able to bring the expression to the form f = f(k) or k = k(f) and then this solution wouldn't work. Is it possible to solve the non-linear expressions with e.g. fsolve or something else within the symbolic expression in such cases?

Mischa Kim 2014-4-14

I don't think I follow. Are you trying to solve a nonlinear equation of the form H(f) = 0 using anonymous functions? If so, simply define H as shown above and then use fsolve(H,f0), where f0 is the starting guess.

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