Intensity and amplitude of a wave - EK question

[Smelt]

p135 in the physics book

As the force with which the hammer strikes the string is increased, the amplitude of the string's motion is increased. Which of the following properties of the wave on the string will remain the same as the force of the hammer is increased?

I. Frequency
II. Wavelength
III. Velocity

Answer is apparently all three will remain the same. How come? Intensity = A^2. And on p124 it states that intensity is dependent on density of the medium, wave freq, and the wave velocity? I'm pretty confused

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[neurodoc]

They are talking about a string of fixed length (as in a piano). If you strike a particular key you get a particular note (a sound of a particular frequency and wavelength). The greater the force with which you strike the key, the louder the sound (because more energy is transferred by the greater displacement (amplitude) of the vibrating spring), but it's still the same note. There may be a little confusion of what "velocity" they are asking about. If we assume they mean the speed of sound in air, that can be considered constant for the purposes of this question. If they really mean the velocity of the string, that can get complicated. You could measure or calculate the motion of various points along the length of the spring while it's vibrating at different amplitudes, but that would be overthinking the problem and is unnecessary. We already know that the frequency of any played piano note doesn't change over a wide range of amplitudes. Thinking in terms of the velocity of the vibrating string itself, we could say that it varies with the frequency...higher frequencies being associated with faster vibrations. However, this problem asks about one specific string of a given length, so even if that is what they mean by velocity, it is still constant.

This is good example of a question in which you can entangled by overthinking, and for which reasoning by analogy can make the problem much simpler to solve.