This allows us to obtain a solution of the wave equation, which gives the below mode shapes for a rectangular membrane of width $\ell_x$ and length $\ell_y$, and which is fixed along all four edges, are given by $$ \phi_{m,n}(x,y) = A_{m,n}\sin\bigg({m \pi \over \ell_x}x\bigg)\sin\bigg({n \pi \over \ell_y} y\bigg), \quad \text{m,n = 1, 2, 3, ..}$$ (where $A_{m,n}$ is a contstant which we set equal to 1). Refer to the lecture notes, which fix the membrane on one of the axes (fixing both axes forces the modes on the membrane to be standing waves in both planes). The mode shape identifier (m,n) refers to the number of antinodes in the $x$ and $y$ directions, respectively. The resonance frequency for the (m,n) mode is $$ f_{m,n}=\omega_{m,n}/2\pi= {c \over 2 \pi} \sqrt{k_x^2+k_y^2} ={c \over 2}\sqrt{\bigg({m \over \ell_x}\bigg)^2 + \bigg({n \over \ell_y}\bigg)^2} \ . $$ It's clear that the frequencies are dependent on the width and length of the membrane. If the dimensions change, then a particular mode denoted by $\phi_{m,n}$ can have a quite different frequency.
Note the modes of 1,4 and 2,2 are degenerate (same frequency). In practise some combination of them will be excited $\phi_{total}= a \phi_{1,4}+ b \phi_{2,2}$, where $a$ and $b$ are constants. See the below examples.
