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 = (𝑇 sin 𝜃𝑥+𝑑𝑥 ) − (𝑇 sin 𝜃𝑥 ) = (𝑇 sin 𝜃𝑥 +
𝜕[𝑇 sin 𝜃]
𝜕[𝑇 sin 𝜃]
𝑑𝑥) − (𝑇 sin 𝜃𝑥 ) =
𝑑𝑥
𝜕𝑥
𝜕𝑥
For small 𝜃 and since 𝑇 is assumed to be constant along the length of the string:
𝜕𝑦
𝜕 2𝑦
sin 𝜃 ≈ tan 𝜃 ≈
→ 𝑑𝐹𝑦 = 𝑇 2 𝑑𝑥
𝜕𝑥
𝜕𝑥
Applying Newton’s second law:
𝑑𝐹𝑦 = (𝑑𝑚)𝑎𝑦
𝑇
𝜕 2𝑦
𝜕 2𝑦
(𝜇𝑑𝑠)
𝑑𝑥
=
𝜕𝑥 2
𝜕𝑡 2
Where 𝜇 is the linear density of the string. For small 𝑑𝑦 (small 𝜃), 𝑑𝑥 ≈ 𝑑𝑠. Thus, the
wave equation is derived:
𝜕 2𝑦 𝑇 𝜕 2𝑦
𝜕 2𝑦
2
=
=
𝑐
𝜕𝑡 2 𝜇 𝜕𝑥 2
𝜕𝑥 2
Components of the simple harmonic solution for a standing wave on a string of length 𝐿
fixed at 𝑥 = 0 and 𝑥 = 𝐿:
𝑦𝑛 (𝑥, 𝑡) = (𝐴𝑛 sin 𝜔𝑛 𝑡 + 𝐵𝑛 cos 𝜔𝑛 𝑡) sin
𝜔𝑛 𝑥
c
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:
∞
∞
𝑦(𝑥, 𝑡) = ∑ 𝑦𝑛 (𝑥, 𝑡) = ∑(𝐴𝑛 sin 𝜔𝑛 𝑡 + 𝐵𝑛 cos 𝜔𝑛 𝑡) sin
𝑛=0
𝑛=0
𝜔𝑛 𝑥
c
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 = 2𝜋 2 𝜌𝑎𝑖𝑟 𝑓𝑣𝑅 2 𝐿 (
1
√2
+
)
𝑀 2𝑀2
Where M is defined as follows:
𝑀=
𝑅 √2𝜋𝑓
2 𝜂𝑎𝑖𝑟
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:
𝜌
2𝑀2
𝜏1 =
(
)
2𝜋𝜌𝑎𝑖𝑟 𝑓 2√2𝑀 + 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 =
𝐸1
𝜋𝑓𝐸2
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 =
1
8𝜇𝐿𝑓 2 𝐺
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.
−1
1
1
1 −1
2𝜋𝜌𝑎𝑖𝑟 𝑓 2√2𝑀 + 1
𝜋𝑓𝐸2
𝜏=[ + + ] =[
(
)
+
+ 8𝜇𝐿𝑓 2 𝐺]
𝜏1 𝜏2 𝜏3
𝜌
2𝑀2
𝐸1
So amplitude decay can be modeled as follows for a particular vibrational frequency:
𝐴 = 𝐴0 𝑒 −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:
𝜕 2 𝑦 𝑇 𝜕 2 𝑦 𝐸𝑆𝐾 2 𝜕 2 𝑦
=
−
𝜕𝑡 2 𝜇 𝜕𝑥 2
𝜇 𝜕𝑥 2
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:
1
𝐾 = √ ∫ 𝑦 2 𝑑𝑆
𝑆
𝑆
Then for a round string of radius R:
𝑅
1
𝐾=√ 2 ∫
𝜋𝑅
√𝑅 2 −𝑦 2
∫
−𝑅 −√𝑅2 −𝑦 2
𝑅
1
1 𝑅4𝜋
𝑅
𝑦 2 𝑑𝑧 𝑑𝑦 = √ 2 ∫ 𝑦 2 (2√𝑅 2 − 𝑦 2 ) 𝑑𝑦 = √ 2 (
)=
𝜋𝑅
𝜋𝑅
4
2
−𝑅
If the stiffness of the string is small, then the result is approximated by scaling all of the
harmonics by the following term:
𝑓𝑛 = 𝑛𝑓0 √1 +
𝜋 2 𝐸𝑆𝐾 2 2
𝜋 3 𝐸𝑅 4 2
√
𝑛
=
𝑛𝑓
1
+
𝑛
0
𝑇𝐿2
4𝑇𝐿2
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.