If the frequency of a periodic wave is cut in half while the speed remains the same, what happens to the wavelength

The wavelength becomes twice the original wavelength

Explanation:

Recall that for regular waves, the relationship between wavelength, velocity (i.e speed) and frequency is given by

v = fλ

where,

v = velocity,

f = frequency

λ = wavelength

Before a change was made to the frequency, we have: v₁ = f₁ λ₁

After a change was made to the frequency, we have: v₂ = f₂ λ₂

We are told that the speed remains the same, so

v₁ = v₂

f₁ λ₁ = f₂ λ₂ (rearranging this)

f₁ / f₂ = λ₂/λ₁ (1)

we are given that the frequency is cut in half.

f₂ = (1/2) f₁     (rearranging this)

f₁/f₂ = 2 (2)

if we substitute equation (2) into equation (1):

f₁ / f₂ = λ₂/λ₁

2 = λ₂/λ₁

λ₂ = 2λ₁

Hence we can see that the wavelength after the change becomes twice (i.e doubles) the initial wavelength.