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Physics Chapter 9 : Topic 11
Harmonics in Stationary Waves
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Harmonics in a stationary wave are produced when:
Reasoning: Harmonics are formed when stationary waves satisfy the required boundary conditions of the medium.
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The lowest possible frequency of vibration of a stretched string is called:
Reasoning: The fundamental frequency is the first and lowest frequency at which a stationary wave is formed.
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The first harmonic of a string fixed at both ends contains:
Reasoning: The fundamental mode has one antinode (one loop) between two fixed ends.
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In the first harmonic of a stretched string fixed at both ends, the length of the string is:
Reasoning: For the fundamental mode, the string length equals half the wavelength.
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The frequency of the second harmonic compared with the fundamental frequency is:
Reasoning: The second harmonic has twice the frequency of the fundamental.
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The third harmonic of a string fixed at both ends has:
Reasoning: The number of antinodes equals the harmonic number.
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For a string fixed at both ends, the wavelength of the nth harmonic is:
Reasoning: For a string, $\mathbf{n\lambda\ /\ 2 = L}$, therefore $\mathbf{\lambda = 2L\ /\ n}$.
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The frequency of $\mathbf{nth}$ harmonic of a stretched string is:
Reasoning: Harmonic frequencies are integer multiples of the fundamental frequency.
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A string vibrating in its second harmonic has:
Reasoning: The second harmonic has two antinodes and three nodes (including both ends).
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The frequency ratio of first, second, and third harmonics is:
Reasoning: Harmonic frequencies are whole number multiples of the fundamental frequency.
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The stationary wave pattern with the greatest number of loops represents:
Reasoning: Higher harmonics contain more loops and have higher frequencies.
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In the fundamental mode of a string fixed at both ends, the ends are:
Reasoning: Fixed ends cannot move, so they always act as nodes.
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The third harmonic wavelength of a string of length $\mathbf{L}$ is:
Reasoning: Using $\mathbf{\lambda}_{\mathbf{n}}\mathbf{= 2L\ /\ n}$, $\mathbf{\lambda}_{\mathbf{3}}\mathbf{= 2L\ /\ 3}$.
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The second harmonic of a string has wavelength:
Reasoning: For second harmonic, $\mathbf{\lambda}_{\mathbf{2}}\mathbf{= 2L\ /\ 2 = L}$.
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When a string vibrates in its fundamental mode, the number of nodes is:
Reasoning: The fundamental mode has nodes only at the two fixed ends.
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The formation of harmonics in a string depends mainly on:
Reasoning: Boundary conditions determine the allowed wavelengths and frequencies.
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If the fundamental frequency of a string is $\mathbf{50\ Hz}$, its third harmonic frequency will be:
Reasoning: Third harmonic frequency $\mathbf{= 3 \times}$ fundamental frequency$\mathbf{= 3 \times 50 = 150\ Hz}$
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A stationary wave on a string contains four loops. This represents:
Reasoning: Each loop corresponds to one antinode, and the number of loops equals harmonic number.
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Which statement about harmonics is correct?
Reasoning: Since frequency increases with harmonic number, wavelength decreases.
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The first overtone of a stretched string corresponds to:
Reasoning: The first overtone is the frequency immediately above the fundamental, which is the second harmonic.
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The second overtone of a string corresponds to:
Reasoning: Fundamental $\mathbf{=}$ first harmonic, first overtone $\mathbf{=}$ second harmonic, second overtone $\ \mathbf{= \ }$ third harmonic.
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A string fixed at both ends can produce:
Reasoning: A string fixed at both ends supports all integer harmonics.
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The energy of vibration is maximum at:
Reasoning: Antinodes have maximum amplitude, so particles possess maximum vibrational energy.
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The distance between two adjacent nodes in the third harmonic is:
Reasoning: Distance between adjacent nodes is always $\mathbf{\lambda\ /\ 2}$.For third harmonic, $\mathbf{\lambda = 2L\ /\ 3\ }$, so $\mathbf{\lambda\ /\ 2 = L\ /\ 3}$.
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A string has length $\mathbf{1\ m}$ and fundamental frequency $\mathbf{100\ Hz}$. The frequency of its fourth harmonic is:
Reasoning: Fourth harmonic frequency $\mathbf{= 4 \times}$ fundamental frequency $\mathbf{= 400\ Hz}$
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In a harmonic mode, the phase difference between two particles separated by one wavelength is:
Reasoning: Points separated by one wavelength are in the same phase, giving a phase difference of $\mathbf{2\pi}$.
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A harmonic is possible on a string when:
Reasoning: Only wavelengths that fit the string length can form stationary wave patterns.
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If the tension of a string is increased, the frequencies of all harmonics:
Reasoning: Wave speed increases with tension, causing all harmonic frequencies to increase.
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The fundamental frequency is also known as:
Reasoning: The fundamental mode is the first harmonic of a vibrating system.
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Harmonics in stationary waves are important because they:
Reasoning: Harmonics represent the allowed vibration frequencies of a system.
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