answer this question of class 11
Answers
Resonances in a wave medium (such as on a string or in the air, for sound) are standing waves; they are analogous to the resonant oscillation of a mass and spring. Unlike the mass and spring which has only a single resonant frequency, a stretched string has many frequencies that are resonant.
orThe notion of resonance can be extended to wave phenomena. Resonances in a wave medium (such as on a string or in the air, for sound) are standing waves; they are analogous to the resonant oscillation of a mass and spring. Unlike the mass and spring which has only a single resonant frequency, a stretched string has many frequencies that are resonant. These frequencies are called the harmonic series and are responsible for the generation of the pleasing tones from a piano, guitar, violin, or other stringed instruments. When we transfer energy to the strings of these instruments, they oscillate at the special frequencies determined by the harmonic series. When we watch the string move when it vibrates at one of the frequencies of the harmonic series, there is a standing wave pattern that is different for each frequency within the harmonic series.
Standing waves are intimately related to the other wave phenomena we have already discussed. We interpret standing waves as a superposition, or sum, of two traveling waves moving in opposite directions along the string. The traveling waves are reflected at the places where the string is firmly held. Since the string is held fixed at the end points, remember that positive wave pulses are reflected back as negative pulses.
Lowest frequency standing wave: fundamental
The lowest standing wave frequency is called the fundamental or first harmonic. For this mode, all parts of the string vibrate together, up and down. Of course, the ends of the string are fixed in place and are not free to move. We call these positions nodes: a node is a point on the string that does not move. As we move along the string, the amplitude of oscillations at each position we look at changes, but the frequency of oscillation is the same. Near a node, the oscillation amplitude is very small.
In the middle of the string, the oscillation amplitude is largest; such a position is defined as an antinode. We assign a wavelength to the fundamental (and each higher harmonic discussed below) standing wave. At a fixed moment in time, all we observe is either a crest or a trough, but we never observe both at the same time for the lowest frequency standing wave. From this, we determine that half a standing wave length fits along the length of the string for the fundamental. Alternatively, we say that the wavelength of the fundamental is twice the length of the string, or
As we'll discuss later, the oscillation frequencies of stretched strings effect the tone of the sounds we hear from instruments such as guitars, violins and cellos. Higher frequency oscillations result in higher-pitched tones; lower frequency oscillations produce lower-pitched tones. So how can we change the oscillation frequency of a stretched string? The above equation tells us. If we either increase the wave speed along the string or decrease the string length, we get higher frequency oscillations for the first (and higher) harmonic. Conversely, reducing the wave speed or increasing the string length lowers the oscillation frequency. How do we change the wave speed? Keep in mind, it is a property of the wave medium, so we have to do something to the string to alter the wave speed. From earlier discussions, we know that tightening the string increases the wave speed. We also know that more massive strings have smaller wave speeds.
As an example of how standing waves on a string lead to musical sounds, consider the first harmonic of a G string on a violin. The string is typically made of nylon, having a density of ~1200 kg/m3. The diameter of the G string is 4 mm. The string is held with a tension of 220 N. The frequency of the first harmonic of the G string is 196 Hz. What is the length of the string?
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