To determine the speed of the transverse wave represented by the equation:
[tex]\[ y(x, t) = 8.0 \sin \left(0.5 \pi x - 4 \pi t - \frac{\pi}{4} \right), \][/tex]
we begin by identifying the standard form of a wave equation. The standard form of a sinusoidal wave traveling along the x-axis is:
[tex]\[ y(x, t) = A \sin (k x - \omega t + \phi), \][/tex]
where:
- [tex]\( A \)[/tex] is the amplitude of the wave,
- [tex]\( k \)[/tex] is the wave number,
- [tex]\( \omega \)[/tex] (omega) is the angular frequency,
- [tex]\( \phi \)[/tex] is the phase constant.
Comparing the given wave equation with the standard form, we can see:
- The amplitude [tex]\( A \)[/tex] is 8.0,
- The wave number [tex]\( k \)[/tex] is [tex]\( 0.5 \pi \)[/tex],
- The angular frequency [tex]\( \omega \)[/tex] is [tex]\( 4 \pi \)[/tex],
- The phase constant [tex]\( \phi \)[/tex] is [tex]\( -\frac{\pi}{4} \)[/tex].
To find the speed [tex]\( v \)[/tex] of the wave, we use the relationship between the angular frequency, wave number, and wave speed:
[tex]\[ v = \frac{\omega}{k}. \][/tex]
Substituting the given values for [tex]\( k \)[/tex] and [tex]\( \omega \)[/tex]:
[tex]\[ k = 0.5 \pi \][/tex]
[tex]\[ \omega = 4 \pi \][/tex]
So the speed [tex]\( v \)[/tex] is:
[tex]\[ v = \frac{\omega}{k} = \frac{4 \pi}{0.5 \pi}. \][/tex]
Simplifying this, we get:
[tex]\[ v = \frac{4 \pi}{0.5 \pi} = \frac{4}{0.5} = 8. \][/tex]
Thus, the speed of the wave is:
[tex]\[ v = 8.0 \, \text{meters per second}. \][/tex]
