To solve this problem, we need to use the relationship between the speed of a wave ([tex]\( v \)[/tex]), its frequency ([tex]\( f \)[/tex]), and its wavelength ([tex]\( \lambda \)[/tex]). This relationship is given by the wave equation:
[tex]\[ v = f \lambda \][/tex]
Here:
- [tex]\( v \)[/tex] is the speed of the wave,
- [tex]\( f \)[/tex] is the frequency,
- [tex]\( \lambda \)[/tex] is the wavelength.
The speed of the wave [tex]\( v \)[/tex] is assumed to be constant in this scenario. To find out what happens to the wavelength when the frequency is multiplied by four, we can manipulate the wave equation.
Initially, let's denote the original frequency by [tex]\( f_1 \)[/tex] and the original wavelength by [tex]\( \lambda_1 \)[/tex]. Thus, the initial wave equation is:
[tex]\[ v = f_1 \lambda_1 \][/tex]
Next, let's denote the new frequency by [tex]\( f_2 \)[/tex]. According to the problem, the new frequency [tex]\( f_2 \)[/tex] is four times the original frequency:
[tex]\[ f_2 = 4 f_1 \][/tex]
We want to find the new wavelength [tex]\( \lambda_2 \)[/tex]. Using the wave equation again with the new frequency and wavelength, we have:
[tex]\[ v = f_2 \lambda_2 \][/tex]
We substitute [tex]\( f_2 \)[/tex] with [tex]\( 4 f_1 \)[/tex]:
[tex]\[ v = (4 f_1) \lambda_2 \][/tex]
Since the speed of the wave ([tex]\( v \)[/tex]) is constant, we can set the two expressions for [tex]\( v \)[/tex] equal to each other:
[tex]\[ f_1 \lambda_1 = 4 f_1 \lambda_2 \][/tex]
To isolate [tex]\( \lambda_2 \)[/tex], we divide both sides of the equation by [tex]\( 4 f_1 \)[/tex]:
[tex]\[ \lambda_2 = \frac{f_1 \lambda_1}{4 f_1} = \frac{\lambda_1}{4} \][/tex]
Thus, the new wavelength [tex]\( \lambda_2 \)[/tex] is one-fourth of the original wavelength [tex]\( \lambda_1 \)[/tex].
To summarize:
When the frequency of a wave is multiplied by four, the wavelength is divided by four (or multiplied by [tex]\( \frac{1}{4} \)[/tex]).
