Question

The wave function of a standing wave is \(y\left( {x,t} \right) = 4.44\;{\rm{mm}}\sin \left[ {\left( {32.5\;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {\rm{m}}}} \right. \\} {\rm{m}}}} \right)x} \right]\sin \left[ {\left( {754\;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {\rm{s}}}} \right. \\} {\rm{s}}}} \right)t} \right]\). For the two traveling waves that make up this standing wave, find the (a) amplitude;

(b) wavelength;

(c) frequency;

(d) wave speed;

(e) wave functions.

(f) From the information given, can you determine which harmonic this is? Explain.

Short answer

(a) The amplitude is, \(2.22\;{\rm{mm}}\).

Step-by-step solution

Identification of the given data

The given data can be listed below as,

The equation is,

\(\left( {x,t} \right) = 4.44\;{\rm{mm}}\sin \left[ {\left( {32.5\;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {\rm{m}}}} \right. \\} {\rm{m}}}} \right)x} \right]\sin \left[ {\left( {754\;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {\rm{s}}}} \right. \\} {\rm{s}}}} \right)t} \right]\) …(1)

Significance of the amplitude

The maximum deviation from equilibrium of a point on a vibrating body or wave in terms of displacement or distance travelled. It is equivalent to the vibration path's half-length.

Determination of the amplitude

The standard equation of the standing wave is expressed as,

\(y\left( {x,t} \right) = \left( {{A_{sw}}\sin kx} \right)\sin \omega t\) …(2)

Here \({A_{sw}}\) is the amplitude of simple harmonic and \(\omega \) is the angular frequency.

But,

\({A_{sw}}=2A\) …(3)

Compare both the equation (1) and (2),

\({A_{sw}} = 4.44\;{\rm{mm}}\)

Substitute the value of \({A_{sw}}\) in the equation (3).

\(\begin{array}{c}2A = 4.44\;{\rm{mm}}\\A = \frac{{4.44\;{\rm{mm}}}}{2}\\A = 2.22\;{\rm{mm}}\end{array}\)

Hence the amplitude is, \(2.22\;{\rm{mm}}\).