string instrument produce sound through

String instruments produce sound through the vibration of taut strings, which generates transverse standing waves that displace air molecules to create pressure waves (sound).

Step 1: A string fixed at both ends vibrates in standing wave patterns when disturbed (plucked, bowed, or struck).
Formula: Wavelength of fundamental mode $\lambda_1 = 2L$, where $L$ is string length (typically $0.6$--$1.0$ m for guitars).

Step 2: Fundamental frequency $f_1$ determines pitch.
Formula: $f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$
$T =$ tension (N), $\mu =$ linear density (kg/m).
Example (guitar E string): $L = 0.65$ m, $T = 74$ N, $\mu = 0.00032$ kg/m.
Substitute: $f_1 = \frac{1}{2(0.65)} \sqrt{\frac{74}{0.00032}} \approx 82$ Hz ($E_2$ note).

Step 3: Harmonics at $f_n = n f_1$ ($n=2,3,\ldots$) add timbre.
Vibrations transfer to bridge $\to$ instrument body $\to$ air (resonance amplifies $\sim 10$--$20$ dB).

Step 4: Sound speed $v = f \lambda \approx 343$ m/s (air at $20^\circ$C).
Waves propagate as longitudinal pressure variations ($\Delta P \sim 10^{-5}$ Pa for audible sound).

Final answer: vibration of strings
vibration of strings