The lossless one-dimension wave equation was previously derived for a string by assuming the restoring force due to string stiffness was negligible. However, it turns out that string stiffness cannot be ignored in many musical contexts.
In general, strings of greater diameter have larger restoring forces due to bending.
This restoring force can be accounted for in the wave equation with a term proportional to the fourth spatial derivative of the string displacement:
where is tension,
is the moment constant for a cylindrical string of radius and Young's modulus .
This equation can be analyzed (see The Dispersive 1D Wave Equation
for details) in order to estimate the resulting frequency-dependent wave velocity as:
where
is the lossless wave velocity.
Higher frequency wave components travel with faster velocities, with the familiar result that the normal modes of the string are no longer perfectly harmonic.
Because dispersion is particularly obvious in piano strings, a stretched tuning system is used in an attempt to minimize beating between simultaneous notes.