1.5 – RMS-Wert eines reinen Tons

 

Berechnen Sie den RMS-Wert des reinen Tons {p^\prime }\left( t \right) = A\sin \left( {\omega t} \right).

Lösung

{p_{rms}} = \sqrt {\frac{1}{T}\int_0^T {{p^\prime }{{\left( t \right)}^2}dt} }

\quad = \sqrt {\frac{1}{T}\int_0^T {{A^2}{{\sin }^2}\left( {\omega t} \right)dt} }

\quad = \frac{A}{{\sqrt T }}{\left( {\left[ {\frac{1}{2}t-\frac{1}{{4\omega }}\sin \left( {2\omega t} \right)} \right]_0^T} \right)^{\frac{1}{2}}}

\quad = \frac{A}{{\sqrt T }}{\left( {\frac{T}{2}-\frac{1}{{4\omega }}\sin \left( {2\frac{{2\pi }}{T}T} \right)} \right)^{\frac{1}{2}}} = \frac{A}{{\sqrt T }}\frac{{\sqrt T }}{{\sqrt 2 }} = \frac{A}{{\sqrt 2 }}