16.14
Modes of Standing Waves - I
A close look at earthquakes provides evidence for the conditions appropriate for resonance, standing waves, and constructive and destructive interfere…
When a taut string is plucked, the resulting waves are non-linear. These non-linear waves are produced by the interference of two waves traveling in opposite directions that have the same frequency and amplitude.
The resulting wave patterns, consisting of nodes and antinodes, are called normal modes or harmonics.
The first harmonic occurs when the wavelength is twice the length of the string, which is also termed the fundamental mode.
The boundary condition is that there are nodes at the two ends of the string. A wave that does not follow this condition is impossible. For waves that follow this condition, twice the length of the string equals the integral multiples of the wavelength.
Based on this, the wavelength of each normal mode can be determined. Recall that wavelength is inversely related to frequency. So, the frequency of each harmonic is the nth multiple of the first harmonic.
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Q1: What causes standing waves to form on a plucked string?
Standing waves form when two waves traveling in opposite directions with the same frequency and amplitude interfere with each other. This interference and superposition of waves creates a pattern of nodes and antinodes along the string. The resulting non-linear wave patterns are called normal modes or harmonics, which remain stationary rather than traveling along the string.
Q2: What are nodes and antinodes in a standing wave?
Nodes are points on the string that remain stationary and never move, while antinodes are points that oscillate with maximum amplitude. In a standing wave on a string, nodes always occur at the fixed ends due to boundary conditions. The alternating pattern of nodes and antinodes characterizes each normal mode or harmonic of the string.
Q3: How does the fundamental mode differ from higher harmonics?
The fundamental mode, or first harmonic, occurs when the wavelength equals twice the string length, creating one antinode between two end nodes. Higher harmonics have shorter wavelengths that fit multiple half-wavelengths along the string length. Each successive harmonic has a frequency that is an integral multiple of the fundamental frequency.
Q4: What boundary condition must be satisfied for standing waves on a string?
The boundary condition requires nodes at both fixed ends of the string. This constraint means the string length must equal an integral multiple of half-wavelengths. Any wave pattern that violates this condition cannot exist on the string, which determines which frequencies can produce standing waves.
Q5: How are harmonic frequencies related to the fundamental frequency?
Each harmonic frequency is an integral multiple of the first harmonic frequency. The second harmonic has twice the frequency of the fundamental, the third harmonic has three times the frequency, and so on. This mathematical relationship arises from the boundary conditions that dictate how wavelengths fit along the string length.
Q6: Why do certain building heights suffer more damage during earthquakes?
Buildings experience resonance when earthquake waves match the building's natural frequency of vibration, causing standing waves to form in the structure. Buildings of specific heights match the boundary conditions for standing waves at those frequencies, leading to constructive interference and amplified oscillations. This resonance effect can devastate buildings of certain heights while leaving neighboring structures intact.
Q7: What determines the possible frequencies of standing waves on a string?
The distance between fixed ends and the propagation speed of the disturbance determine possible standing wave frequencies. The symmetrical boundary conditions—nodes at each end—restrict which wavelengths can fit on the string. Only frequencies that allow integral multiples of half-wavelengths to fit between the fixed ends can excite standing waves.