To determine what happens to the wavelength of a wave when its speed is quadrupled while the frequency is doubled, we can use the equation that relates the speed, frequency, and wavelength of a wave.
The equation is:
v = λf
where v represents the speed of the wave, λ represents the wavelength, and f represents the frequency.
Let's consider the initial values:
Speed (v₁) = Initial speed Frequency (f₁) = Initial frequency Wavelength (λ₁) = Initial wavelength
And the final values after the changes:
Speed (v₂) = Quadrupled speed (4 times the initial speed) Frequency (f₂) = Doubled frequency (2 times the initial frequency) Wavelength (λ₂) = Unknown (what we're trying to determine)
Using the equation, we have:
v₁ = λ₁f₁ (Initial equation) v₂ = λ₂f₂ (Final equation)
Since the frequency is doubled (f₂ = 2f₁) and the speed is quadrupled (v₂ = 4v₁), we can rewrite the final equation as:
4v₁ = λ₂(2f₁)
Dividing both sides of the equation by 2f₁:
2v₁ / f₁ = λ₂
Now, let's substitute the expression for speed (v) in terms of wavelength (λ) and frequency (f) from the initial equation:
2(λ₁f₁) / f₁ = λ₂
Simplifying:
2λ₁ = λ₂
Therefore, when the speed is quadrupled while the frequency is doubled, the wavelength remains unchanged. The wavelength of the wave remains the same in this scenario.