Answered

What are the wavelength (in meters) and frequency (in hertz) of a wave whose displacement is given by the equation [tex] y = 0.5 \sin (0.20 x + 120 t) [/tex], where [tex] x [/tex] and [tex] y [/tex] are in meters and [tex] t [/tex] is in seconds?

A. [tex] 5.0 \, \text{m}, \, 10 \, \text{Hz} [/tex]
B. [tex] 19 \, \text{m}, \, 120 \, \text{Hz} [/tex]
C. [tex] 0.20 \, \text{m}, \, 120 \sqrt{2 \pi} \, \text{Hz} [/tex]
D. [tex] 10 \, \text{m}, \, 0.50 \, \text{Hz} [/tex]
E. [tex] 31 \, \text{m}, \, 19 \, \text{Hz} [/tex]



Answer :

GinnyAnswer
To determine the wavelength and frequency of a wave given by the displacement equation [tex]\( y = 0.5 \sin (0.20 x + 120 t) \)[/tex], we need to identify the wave number (k) and the angular frequency ([tex]\(\omega\)[/tex]) from the equation.

The general form of a wave equation is:
[tex]\[ y = A \sin (kx + \omega t + \phi) \][/tex]
where:
- [tex]\( A \)[/tex] is the amplitude of the wave,
- [tex]\( k \)[/tex] is the wave number,
- [tex]\( \omega \)[/tex] is the angular frequency,
- [tex]\( x \)[/tex] is the position,
- [tex]\( t \)[/tex] is the time,
- [tex]\( \phi \)[/tex] is the phase constant.

From the given equation:
[tex]\[ y = 0.5 \sin (0.20 x + 120 t) \][/tex]

we can immediately see that:
- The coefficient of [tex]\( x \)[/tex] is [tex]\( k = 0.20 \)[/tex] radians per meter,
- The coefficient of [tex]\( t \)[/tex] is [tex]\( \omega = 120 \)[/tex] radians per second.

### Calculation of Wavelength

The wavelength [tex]\( \lambda \)[/tex] is related to the wave number [tex]\( k \)[/tex] by the formula:
[tex]\[ \lambda = \frac{2\pi}{k} \][/tex]

Substituting the value of [tex]\( k \)[/tex]:
[tex]\[ \lambda = \frac{2\pi}{0.20} \][/tex]
[tex]\[ \lambda = \frac{2\pi}{0.20} = \frac{2\pi}{\frac{1}{5}} = 2\pi \times 5 = 10\pi \][/tex]
[tex]\[ \lambda = 10\pi \approx 31.415 \, \text{meters} \][/tex]

Thus, the wavelength of the wave is approximately [tex]\( 31 \)[/tex] meters.

### Calculation of Frequency

The frequency [tex]\( f \)[/tex] is related to the angular frequency [tex]\( \omega \)[/tex] by the formula:
[tex]\[ f = \frac{\omega}{2\pi} \][/tex]

Substituting the value of [tex]\( \omega \)[/tex]:
[tex]\[ f = \frac{120}{2\pi} \][/tex]
[tex]\[ f = \frac{120}{2\pi} = \frac{120}{6.2832} \approx 19.098 \, \text{hertz} \][/tex]

Thus, the frequency of the wave is approximately [tex]\( 19 \)[/tex] hertz.

Based on these calculations, the correct answer is:

[tex]\[ \boxed{31 \, \text{m}, 19 \, \text{Hz}} \][/tex]