(II) On an electric guitar, a "pickup" under each string...

(II) On an electric guitar, a "pickup" under each string transforms the string's vibrations directly into an electrical signal. If a pickup is placed 16.25 $\mathrm{cm}$ from one of the fixed ends of a 65.00 -cm-long string, which of the harmonics from $n=1$ to $n=12$ will not be "picked up" by this pickup?

Key Concepts

Nodal Points in Vibrating Strings

Nodal points are specific locations along a vibrating string where the displacement is always zero during its motion. These points occur naturally in standing wave patterns and depend on the harmonic being excited. The presence of a node at a particular position means that at that location, the string’s vibration amplitude is zero.

Impact of Pickup Position on Signal Detection

In musical instruments like the electric guitar, pickups convert the vibrations of a string into an electrical signal. If a pickup is positioned at a location that coincides with a nodal point for a particular harmonic, it will not detect that harmonic because the displacement of the string is zero at that point. Thus, the placement of the pickup relative to the nodes affects which harmonics are effectively picked up and amplified.

Standing Waves on a String

Standing waves occur when two waves of the same frequency and amplitude travel in opposite directions along a string and interfere. This interference produces fixed vibration patterns, where certain positions along the string, called nodes, remain stationary, and other positions, called antinodes, exhibit maximum vibration.

Harmonics and Mode Shapes

Harmonics are the natural frequencies at which a string vibrates, with each harmonic corresponding to a specific standing wave pattern or mode shape. The fundamental harmonic (first harmonic) involves a half wavelength along the string, and higher harmonics contain additional nodes and antinodes, dividing the string into an increasing number of equal segments.

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