I’m currently going through harmonics, and I do not at all understand the fundamental frequency. I understand that it is the simplest vibration of a string, but I don't understand how can it have frequency at all if it is only half a wavelength. Isn’t frequency how many cycles are completed per second, and isn’t the fundamental frequency only half a cycle if it is half a wavelength? How can there be frequency of (say) 162 cycles per second if one cycle doesn’t even complete in the medium of the string? Is it measuring the frequency of the half wavelength as a full cycle? Is there frequency measured as a whole cycle from one half because it is a resultant wave of two waves making the half wavelength? If so, why do we have to multiply by two to get the wavelength of the cycle from the string length?
[Physics] fundamental frequency, how does it make sense
acousticsfrequencyharmonicsstringwavelength
Best Answer
When a string, fixed a both ends, vibrates in the fundamental mode, the perpendicular displacement $\phi_1(x,t)$ of a point located at $x$ along the length of the string is given by
$$\phi_1(x,t) = A_1(t)\phi_1(x) = A_1\cos(2\pi f_1t + \varphi)\sin\left(\frac{\pi}{L}x\right)$$
Now, it is true that the spatial variation of the fundamental mode is a 'half-cycle' since the argument of the $\sin$ ranges from $0$ to $\pi$.
However, the fundamental frequency refers to the time dependent amplitude $A_1(t)$. Note that $A_1(t)$ executes $f_1$ cycles per second. Take a look at this animated gif of the fundamental mode and the first three harmonics:
Animated gif credit
See that although the fundamental mode has a 'half-cycle' spatial variation, the time dependent amplitude goes from a maximum, through zero, a minimum, back through zero back to the maximum in a time $T_1 = \frac{1}{f_1}$ where $f_1$ is the fundamental frequency.
Also note that the frequency of the 2nd harmonic is twice the frequency of the fundamental, the frequency of the 3rd harmonic is thrice the frequency of the fundamental and so on.