When you strike a string harder, it does indeed result in a larger amplitude of vibration. The amplitude of vibration is directly proportional to the energy imparted to the string. A harder strike means more energy is transferred to the string, causing it to vibrate with greater displacement from its resting position.
However, the pitch of a sound, which determines whether it is higher or lower, is primarily determined by the frequency of vibration, not the amplitude. The frequency is the number of complete cycles of vibration the string completes in one second and is measured in Hertz (Hz).
In a well-tuned musical instrument like a guitar or piano, when you strike a string harder, the increase in energy results in a larger amplitude of vibration but does not significantly affect the tension, mass, or length of the string. Therefore, the fundamental frequency (the main pitch produced) remains the same, and the note remains at the same pitch.
For example, if a guitar string is tuned to produce the note A (440 Hz) and you pluck it softly or strum it with more force, the amplitude of the vibration will vary, but the fundamental frequency will still be 440 Hz, and you will hear the same pitch of A.
However, it's worth noting that increasing the amplitude can introduce additional overtones or harmonics. Harmonics are integer multiples of the fundamental frequency that contribute to the timbre or tonal quality of the sound. So, while the pitch remains the same, the sound might be perceived as "richer" or "fuller" due to the presence of more pronounced harmonics caused by the increased amplitude.
In summary, striking a string harder increases the amplitude of vibration and can affect the timbre of the sound by introducing more harmonics, but it does not change the fundamental frequency or pitch of the note produced. Pitch is primarily determined by the frequency of vibration, which depends on the tension, mass, and length of the vibrating medium (in this case, the string).