The author develops the Trefftz method using piecewise particular solutions for solving the Helmholtz equation \(\Delta u+k^{2}u=f\) where \(k^{2}\) is not exactly equal, but may be very close, to an eigenvalue of the operator -\(\Delta \), derives a new error analysis, which reveals the error bounds, particularly at the degeneracy case when \(k^2=\lambda _{l}\). The advantages of the Trefftz method are high accuracy of the Helmholtz solutions with exponential convergence rates. The paper may be regarded as a development of some papers of the same author where the Trefftz method is used for the Laplace equation.