The relationship between the amplitude, wavelength, and intensity of a sound wave is governed by the physics of wave propagation. Let's break down the two scenarios you mentioned:
Doubling the amplitude: The intensity of a sound wave is directly proportional to the square of its amplitude. When you double the amplitude of a sound wave, you are effectively increasing it by a factor of 2. Since intensity is proportional to the square of the amplitude, doubling the amplitude will result in an intensity increase by a factor of 2^2 = 4. This means that the intensity will quadruple.
Halving the wavelength: The intensity of a sound wave is inversely proportional to the square of its wavelength. When you halve the wavelength of a sound wave, you are effectively reducing it by a factor of 1/2. Since intensity is inversely proportional to the square of the wavelength, halving the wavelength will result in an intensity decrease by a factor of (1/2)^2 = 1/4. This means that the intensity will be reduced to one-fourth of its original value, i.e., halved.
In summary, the relationship between amplitude and intensity is quadratic, while the relationship between wavelength and intensity is inverse quadratic. Doubling the amplitude leads to a quadrupling of intensity because of the squared relationship, while halving the wavelength only halves the intensity due to the inverse squared relationship.