Mosis said:oh, I think I figured it out
the free end of a string will always be an anti-node, and a wave is always flat at an anti-node, so we must always have the spatial derivative = 0 at the end (and at any antinode, for that matter)
The fundamental mode of an open-end string refers to the lowest resonant frequency that can be produced by the string when it is plucked or struck. This mode is also known as the first harmonic and is the most basic and purest tone that a string can produce.
The fundamental mode of an open-end string is important because it determines the pitch or note that the string will produce. It is the basis for all other harmonics and overtones that can be produced by the string, making it a crucial aspect of stringed instruments and music in general.
The fundamental mode of an open-end string is determined by several factors, including the length, tension, and mass of the string. These factors affect the speed at which the string vibrates and the wavelength of the sound produced, which ultimately determines the pitch or frequency of the fundamental mode.
Yes, the fundamental mode of an open-end string can be changed by altering the length, tension, or mass of the string. For example, plucking a shorter section of the string will produce a higher frequency and pitch, while increasing the tension or mass of the string will lower the frequency and pitch of the fundamental mode.
The fundamental mode of an open-end string serves as the basis for all other modes and overtones that can be produced by the string. These modes are integer multiples of the fundamental frequency, creating a harmonic series of tones. The fundamental mode is also the loudest and most prominent of these modes, with the other modes and overtones adding complexity and richness to the overall sound of the string.