Spring-supported acoustic membrane wave

For a massless spring-supported membrane adjoining a fluid half-space, kinematic and pressure balance give the dispersion relation $A{s}^4+s^2-1=0$ with $s=c/c_0$.

For $A{s}^4+s^2-1=0$, the physical speed approaches $c_0(1-A/2)$ as $A$ tends to zero and scales as $c_0{A}^{-1/4}$ as $A$ tends to infinity.