Theory—

In order to anticipate the acoustic behavior of the bass guitar and clarify its analysis, it is useful to construct a physical model of the system. The quality of the tone produced by the instrument is assessed by examining the harmonic content of the signal delivered by the output jack. This signal is a product of the construction of the instrument and the materials employed. Initially, the audible waveform is produced mechanically by the vibrational motion of the string in response to being plucked. Two permanent magnet pickups then independently convert the vibration of the metal string into electrical signals, which are combined and output through the jack. The string is a soft magnet and its motion modulates the magnetic flux through coils of wire in the pickups which produces an alternating current in the wire. The goal of this theoretical model will be to predict the motion of the segments of string that are directly above the pickups, for the electrical output is derived only from this information.

Transverse Vibration of a Fixed-Fixed String—

The first approach to modeling the vibrational motion of a bass string-body system assumes that the string is pinned to fixed supports of infinite mass. Also the string is taken to have zero stiffness, and behave perfectly elastically. The tension is assumed to be constant throughout the string (even though a real string would have longitudinal vibration as well). For now, all damping mechanisms are neglected.

Expanding the differential force balance in the y-direction:

dFy=Tsinθx+dx-Tsinθx=Tsinθx+∂Tsinθ∂xdx-Tsinθx=∂Tsinθ∂xdx

For small θ and since T is assumed to be constant along the length of the string:

sinθ≈tanθ≈∂y∂x → dFy=T∂2y∂x2dx

Applying Newton’s second law:

dFy=dmay

T∂2y∂x2dx=μds∂2y∂t2

Where μ is the linear density of the string. For small dy (small θ), dx≈ds. Thus, the wave equation is derived:

∂2y∂t2=Tμ∂2y∂x2=c2∂2y∂x2

Components of the simple harmonic solution for a standing wave on a string of length L fixed at x=0 and x=L:

ynx,t=Ansinωnt+Bncosωntsinωnxc

With ωn=nπc/L. Thus the general solution for the motion of a fixed-fixed string is the sum of all of the harmonic modes:

yx,t=n=0∞ynx,t=n=0∞Ansinωnt+Bncosωntsinωnxc

String-Body System—

Damping—

Three factors contribute to damping the vibration of a tensioned string: viscous drag in air, internal damping, and energy loss through the supports. These phenomena remove energy from the vibrational modes of the string over time. Thus, the harmonic content of a note is time dependent, and a guitar can only sustain vibration above some threshold amplitude (and ultimately some volume) for a particular length of time.

The viscous retarding force on a round vibrating string in air with radius R and length L is given by the following equation:

Fr=2π2ρairfvR2L2M+12M2

Where M is defined as follows:

M=R22πfηair

Here, ρair is the density of air, ηair is the kinematic viscosity of air, v is the velocity of the string, and f is the frequency of vibration. This gives an exponential decay with a time constant:

τ1=ρ2πρairf2M222M+1

Internal damping in the material of the sting arises from the fact that a strain is not produced instantaneously in response to a stress in the string. The real response is characterized by some initial strain followed by a slight increase over some time. The phenomenon can be respresented by employing a complex modulus of elasticity:

E=E1+jE2

Internal damping is generally negligible for the solid metal strings used in a bass guitar but contributes greatly to the damping of gut or nylon string vibrations used in other instruments. This gives a decay time constant:

τ2=E1πfE2

When a bass string vibrates, some of the energy is transferred to the support since it is not perfectly stationary or infinitely massive. The associated time constant for the exponential decay is:

τ3=18μLf2G

Here, G is the mechanical conductance of the support material.

Taking all of these damping mechanisms simultaneously will produce a total time decay constant that is the reciprocal of the sum of the reciprocals of the others.

τ=1τ1+1τ2+1τ3-1=2πρairfρ22M+12M2+πfE2E1+8μLf2G-1

So amplitude decay can be modeled as follows for a particular vibrational frequency:

A=A0e-τt

Stiff String—

Real bass guitar strings have some nonzero bending stiffness, which contributes to a small extent to the restoring force that produces vibration motion. Bending stiffness is incorporated intot he wave equation by adding a term as follows:

∂2y∂t2=Tμ∂2y∂x2-ESK2μ∂2y∂x2

Here, E is the modulus of elasticity of the string, S is the cross-sectional area, and K is the radius of gyration, which is found by dividing the second moment of inertia by S and taking the square root:

K=1SS y2dS

Then for a round string of radius R:

K=1πR2-RR-R2-y2R2-y2y2 dz dy=1πR2-RRy22R2-y2 dy=1πR2R4π4=R2

If the stiffness of the string is small, then the result is approximated by scaling all of the harmonics by the following term:

fn=nf01+π2ESK2TL2n2=nf01+π3ER44TL2n2

Here, f0 is the fundamental frequency of the string without taking into account stiffness. Thus, all of the harmonics are slightly sharp. Even though the higher harmonics are affected more, they have less energy; therefore, it is hard to notice this phenomenon. This behavior comes into play when tuning piano strings, but is essentially negligible in the case of a bass guitar.

Dispersion Forces—

Waves on a string with nonzero stiffness and mass have a velocity that is dependent on frequency. The result is that a pulse traveling along a real string changes shape since different harmonics do not move at the same speed relative to each other. The group velocity differs from the crest velocity because the relationship between wave number and frequency is no longer linear. Ultimately, there exists some maximum frequency fc above which a wave cannot propagate on a real string. This means a finite number of harmonics are produced when any particular note is played on a bass guitar.

Torsional Vibrations—

A torque about the axis of the string is produced when it is plucked or struck. This produces another mode of vibration with a net restoring torque dependent on the shear modulus of the material. Torsional vibrations due occur on the strings of a bass guitar, but do not significantly contribute to the sound.

References—

Fletcher, Neville, and Thomas Rossing. The Physics of Musical Instruments. New York: Springer, 1998.