To find the mathematical expression for the wave, we need to use the wave equation:
[tex]\[ y(x, t) = A \sin(kx - \omega t + \phi) \][/tex]
where:
- [tex]\( A \)[/tex] is the amplitude
- [tex]\( k \)[/tex] is the angular wave number
- [tex]\( \omega \)[/tex] is the angular frequency
- [tex]\( \phi \)[/tex] is the phase constant
- [tex]\( x \)[/tex] and [tex]\( t \)[/tex] are the position and time variables, respectively
Given the properties of the wave:
- Wavelength [tex]\( \lambda = 0.37 \)[/tex] meters
- Period [tex]\( T = 0.77 \)[/tex] seconds
- Wave speed [tex]\( v = 12 \)[/tex] meters/second
Step-by-Step Solution:
1. Calculate the Angular Wave Number ([tex]\( k \)[/tex]):
The angular wave number [tex]\( k \)[/tex] is given by:
[tex]\[
k = \frac{2\pi}{\lambda}
\][/tex]
Substituting the given wavelength ([tex]\( \lambda = 0.37 \)[/tex] meters):
[tex]\[
k = \frac{2\pi}{0.37} \approx 16.98 \, \text{m}^{-1}
\][/tex]
2. Calculate the Angular Frequency ([tex]\( \omega \)[/tex]):
The angular frequency [tex]\( \omega \)[/tex] is given by:
[tex]\[
\omega = \frac{2\pi}{T}
\][/tex]
Substituting the given period ([tex]\( T = 0.77 \)[/tex] seconds):
[tex]\[
\omega = \frac{2\pi}{0.77} \approx 8.16 \, \text{s}^{-1}
\][/tex]
3. Assume the Amplitude ([tex]\( A \)[/tex]) and Phase Constant ([tex]\( \phi \)[/tex]):
In many cases, if the amplitude and phase constant are not provided, we assume a unit amplitude for simplicity ([tex]\( A = 1 \)[/tex]) and a zero phase constant ([tex]\( \phi = 0 \)[/tex]). This assumption simplifies the equation.
4. Construct the Wave Equation:
Given that the wave is traveling in the negative [tex]\( x \)[/tex]-direction, the mathematical expression of the wave takes the form [tex]\( y(x, t) = A \sin(kx - \omega t + \phi) \)[/tex]. With the assumptions and calculated values:
[tex]\[
y(x, t) = 1 \sin(16.98x - 8.16t + 0)
\][/tex]
Thus, the mathematical expression for the wave is:
[tex]\[ y(x, t) = 1 \sin(16.98x - 8.16t) \][/tex]
