Resonance

When an external force that oscillates back and forth is applied to the weight of a pendulum, if the frequency of the external force matches the natural frequency of the pendulum (the frequency when resistance is zero), the pendulum will swing vigorously. This is called resonance.

There is a story that Masashige Kusunoki, whose childhood name was Tamonmaru, repeatedly pushed a heavy bell that swayed gently back and forth with his finger every time it moved backward, and eventually managed to swing the bell farther. This is an application of resonance. The average work done by the external force of resonance on a mass point per unit time (this work is equal to the energy absorbed by the mass point per unit time) is maximized when the frequency of the external force is equal to the natural frequency of the mass point. If you hold the top end of a simple pendulum string and swing it back and forth at a frequency equal to the pendulum's natural frequency, the weight of the pendulum will swing vigorously even though the amplitude of the displacement of the top end of the string is small. In this case, the coordinate system that oscillates back and forth together with the top end of the string has an acceleration relative to the inertial system, and as a result, an oscillating external force acts on the weight of the pendulum. If the frequency of the external force is within a certain frequency range centered on the pendulum's natural frequency, the pendulum will swing quite vigorously. The width of this frequency range is called the resonance width. In other words, a curve drawn with the frequency of the external force on the horizontal axis and the average work that the external force does on a mass point per unit time on the vertical axis is called a resonance curve. The maximum value of this curve is where the frequency coincides with the natural frequency, and the difference between the frequency that gives half of this maximum value and the natural frequency is called the half-width of the resonance curve. When the resistive force acting on a mass point is proportional to the speed of the mass point, this proportionality coefficient is proportional to the half-width of the resonance curve. In other words, the smaller the proportionality coefficient of the resistive force to the speed, the sharper the peak of the resonance curve. When resonance occurs, the frequency of the external force that maximizes the amplitude of the vibration of the mass point is somewhat lower than the natural frequency of the mass point.

If the prongs of a vibrating tuning fork are placed close to the opening of a tube with a bottom and water is poured into the tube, the air column inside the tube will emit a strong sound when it reaches a certain length. At this time, the air column resonates with the vibration of the tuning fork and emits sound waves to the outside. Tuning forks are often used with their handles attached to a resonance box. The natural frequency of the air column inside the resonance box is almost the same as the fundamental frequency of the tuning fork. However, this air column does not have a natural frequency that matches the overtones of the tuning fork. Therefore, the air column inside the resonance box resonates with the fundamental vibration of the tuning fork, and a pure, strong sound of this frequency is emitted to the outside. If a coil L, a capacitor C, and an electrical resistance R are connected in series and connected to an AC power source (such as a low-frequency oscillator), when the frequency of the AC power source matches the natural frequency of a circuit with zero resistance R (an LC circuit), a maximum current flows through the circuit. This is resonance in electrical vibration.

[Yoshiro Kainuma]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

振り子のおもりに左右に振動する外力を加えた場合に、外力の振動数が振り子の固有振動数（抵抗ゼロの場合の振動数）に一致すれば振り子は勢いよく振れる。これを共振または共鳴という。

　幼名を多聞丸と称した少年の日の楠木正成(くすのきまさしげ)は、かすかに揺れる重い釣鐘が後退するたびごとに指で押すことを繰り返し、ついには大きく鐘を揺り動かすことができたという話が伝わっているが、これはこの共振を応用したものである。共振の外力が質点に単位時間にする仕事（この仕事は質点に単位時間に吸収されるエネルギーに等しい）の平均値は、外力の振動数が質点の固有振動数と一致するときに最大になる。単振り子の糸の上端を持って、振り子の固有振動数に等しい振動数で左右に振れば、糸の上端の変位の振幅は小さくても、振り子のおもりは勢いよく振れる。この場合には糸の上端とともに左右に振動する座標系は慣性系に対して加速度をもつので、その結果として、振り子のおもりには振動的な見かけの外力が働くことになるのである。外力の振動数が振り子の固有振動数を中心とする一定の幅の振動数領域のなかにあれば、振り子はかなり勢いよく振れる。この振動数領域の幅を共振の幅という。すなわち外力の振動数を横軸にとり、外力が質点に単位時間にする仕事の平均値を縦軸にとって描いた曲線を共振曲線とよぶ。この曲線の最大値は、振動数が固有振動数に一致するところにあるが、この最大値の半分の値を与える振動数と固有振動数との差を共振曲線の半減幅という。質点に働く抵抗力が質点の速度に比例する場合には、この比例係数は共振曲線の半減幅に比例する。すなわち、抵抗力の速度に対する比例係数が小さいほど、共振曲線は鋭いピークをもつ。共振がおこるとき、質点の振動の振幅を最大にするような外力の振動数は、質点の固有振動数よりもいくらか低い値をとる。

　底のある管の口に振動している音叉(おんさ)の脚を近づけておき、管の中に水を注いでいくと、管内の空気柱はある長さになったときに強い音を発する。このとき、空気柱は音叉の振動に共振し、音波を外部に放出するのである。音叉はその柄を共鳴箱につけて使用することが多い。共鳴箱の中の空気柱の固有振動数は音叉の基本振動数にほぼ一致する。しかしこの空気柱は音叉の倍振動と一致する固有振動数をもたない。それで、音叉の基本振動に共鳴箱の中の空気柱が共振し、この振動数の純粋な強い音が外部へ放出されることになる。コイルL、コンデンサーC、電気抵抗Rを直列につなぎ、交流電源（低周波発振器など）に接続した場合、交流電源の振動数が抵抗Rがゼロの回路（LC回路）の固有振動数と一致するとき、回路には最大の電流が流れる。これは電気振動における共振である。

［飼沼芳郎］

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