We consider application of the discrete gradient method for the Webster equation, which models sound waves in tubes. Typically Hamilton equations are described by the use of gradients of the Hamiltonian and it is indispensable to introduce an inner product to define a gradient. We first apply the discrete gradient method to design an energy-preserving method by using a weighted inner product. Comparing with another scheme that is derived by a standard inner product, we show that the discrete gradient method has a geometric invariance, which implies that the method reflects the symplectic geometric aspect of mechanics.