固有振動
∂ 2 y ∂ x 2 = 1 v 2 ∂ 2 y ∂ t 2 {\displaystyle {\frac {{\partial }^{2}y}{\partial x^{2}}}={1 \over {v^{2}}}{\frac {{\partial }^{2}y}{\partial t^{2}}}}
の波動方程式を得る。この波動方程式を解くと、
y n ( x , t ) = A n sin  ( 2 n − 1 ) π x 2 l sin  ( ω n t + φ n ) ( n = 1 , 2 , 3 , ... ) {\displaystyle y_{n}(x,t)=A_{n}\sin {(2n-1)\pi x \over 2l}\sin({\omega _{n}t+\phi _{n}})\quad (n=1,2,3,\ldots )}