In 2 D a string becomes a drum.  Here are the modes of a 2 d drum.

Lx = 1.5;
Ly = 1;
c = 1;
d[x_,y_,m_,n_] = Sin[m Pi x/Lx ]Sin[n Pi y/Ly];
w[n_,m_] = c Sqrt[(m Pi/Lx)^2 + (n Pi/Ly)^2];

ContourPlot[d[x,y,1,1],{x,0,Lx},{y,0,Ly}]

ContourPlot[d[x,y,2,1],{x,0,Lx},{y,0,Ly}]

ContourPlot[d[x,y,1,2],{x,0,Lx},{y,0,Ly}]

w[1,2]
w[2,1]

However, now suppose the drum is square

Lx = 1;
Ly = 1;
c = 1;
d[x_,y_,m_,n_] = Sin[m Pi x/Lx ]Sin[n Pi y/Ly];
w[n_,m_] = c Sqrt[(m Pi/Lx)^2 + (n Pi/Ly)^2];

w[1,2]
w[2,1]