Vibration of a membrane strip with a segment of higher density: analysis of trapped modes

Abstract

The Helmholtz equation governs the dynamics of membranes and waveguides. Using the mode matching method, trapped modes are found for an infinite strip with a segment of inhomogeneity. The exact frequencies and mode shapes are determined as a function of density ratio and length of the segment. Nonuniqueness and nonexistence are demonstrated.

Appendix

That the eigenvalues of the present problem are real is briefly shown as follows. Combine the three regions and regard k complex, $\lambda$ positive and piecewise continuous, and w complex and continuous. The Helmholtz equation is

$$ \nabla^{2} w + \lambda k^{2} w = 0 $$

(20)

Taking the complex conjugate (denoted by an asterisk) gives

$$ \nabla^{2} w^{*} + \lambda (k^{2} )^{*}w^{*} = 0 $$

(21)

Construct

$$ \iint {(w^{*}\nabla^{2} w - w\nabla^{2} w^{*})d\varOmega = [(k}^{2} )^{*} - k^{2} ]\iint {\lambda w^{*}wd\varOmega } $$

(22)

where the integrations are over the entire strip$\varOmega$. Using Green’s identity the left hand side is

$$ \oint {\left( {w^{*}\frac{dw}{dn} - w\frac{dw^{*}}{dn}} \right)} dS = 0 $$

(23)

where n is the normal and S is the boundary. Eq.(23) is zero because both w and w* are zero on sides and at infinity due to the decay of a trapped mode. Since $\lambda w^{*}w$ is positive, $k^{2}$ must be real. Since k cannot be purely imaginary for decay, k must be real.

This argument does not apply to the Helmholtz equation with damping or radiation conditions where the eigenvalues may be complex [20, 21].