The intensity of a wave is directly proportional to the square of its amplitude. Since the two waves have the same amplitude, their intensities will be directly proportional to the square of their amplitudes.
Let's denote the amplitude of the first wave as A₁ and the amplitude of the second wave as A₂. Given that the first wave has twice the wavelength of the second wave, we can write:
λ₁ = 2λ₂
The relationship between wavelength and amplitude in a wave is given by:
λ = c/f
where λ is the wavelength, c is the speed of the wave, and f is the frequency. Since the waves travel at the same speed, their frequencies will be inversely proportional to their wavelengths:
f₁ = c/λ₁ f₂ = c/λ₂
Using the relationship between wavelength and frequency, we can write:
λ₁/λ₂ = f₂/f₁
Substituting the values of λ₁ and λ₂, we get:
2λ₂/λ₂ = f₂/f₁ 2 = f₂/f₁
Since the frequencies are directly proportional to the square roots of the intensities, we can write:
√I₂/√I₁ = 2
Squaring both sides of the equation, we get:
I₂/I₁ = (2)² I₂/I₁ = 4
Therefore, the ratio of the intensities of the two waves is 4:1.