16.4
波動関数のグラフ化
正x方向に移動する正弦波の波動方程式を考えてみましょう。 波動方程式は位置と時間の両方の関数です。 波動方程式から、2 つの異なるグラフをプロットできます。
特定の時間が取られた場合、たとえば t = 0、それは「スナップショット」を意味します。 波形の を取得し、得られたグラフは t=0 での波形…
x方向に進行する正弦波を考えてみましょう。
その波動方程式は変位と時間の関数であるため、媒体内の粒子の動きは、変位位置グラフと変位時間グラフでグラフィカルに表すことができます。
一定の時間では、パーティクルの変位は位置の関数として変化します。
これは、粒子が平衡位置から変位することを表します。その後、このグラフから波長を推測できます。
弦の横波の場合を考えると、グラフは特定の瞬間における弦の実際の形状を表しています。
特定の座標を選択すると、波の方程式をグラフ化すると、変位-時間グラフが作成されます。
このグラフを使用して、周期(波が1つの波長を移動するのに必要な時間)が推定されます。
波動方程式では、コサイン関数の引数は波の位相と呼ばれます。
位相速度は、位相を一定に保ちながら波が移動する速度です。
時間に関して微分を取ると、位相速度の式が得られます。
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Q1: What does a displacement versus position graph show for a wave?
A displacement versus position graph represents a snapshot of the wave at a fixed time, showing how particle displacement varies with position. For a transverse wave on a string, this graph depicts the actual shape of the string at that instant. The wavelength can be determined by measuring the distance between two consecutive crests or troughs with the same height and slope.
Q2: How do you find the period of a wave from a displacement-time graph?
A displacement-time graph shows particle displacement as a function of time at a specific position. The period is the time required for one complete oscillation of the particle. By observing the graph, you can identify the time interval between successive identical points in the wave cycle to determine the period.
Q3: What is the phase of a wave and what does it represent?
The phase is the argument of the cosine function in the wave equation, measured in radians. It determines which part of the sinusoidal cycle is occurring at a particular point and time. For a crest, the phase equals 0, 2π, 4π, etc.; for a trough, the phase equals π, 3π, 5π, etc.
Q4: What is phase velocity and how does it relate to wave motion?
Phase velocity is the speed at which a wave moves while keeping the phase constant. It describes how fast the wave pattern propagates through the medium. The phase velocity can be derived by taking the derivative of the wave equation with respect to time, providing a mathematical expression for wave propagation speed.
Q5: How are crests and troughs identified on a wave graph?
The crest is the highest point of the wave above equilibrium, while the trough is the lowest point below equilibrium. On a displacement versus position graph, crests appear as maximum positive displacements and troughs as maximum negative displacements. The distance between consecutive crests or troughs with identical height and slope equals the wavelength.
Q6: Why are two different graphs needed to fully describe a sinusoidal wave?
A sinusoidal wave depends on both position and time. The displacement versus position graph shows the wave's spatial shape at one instant, revealing wavelength. The displacement versus time graph shows how a single particle oscillates, revealing the period. Together, these graphs provide complete information about wave behavior in space and time.
Q7: How does the wave equation relate to graphical representations of waves?
The wave equation is a mathematical function of both position and time. By fixing time, you obtain a displacement versus position graph showing spatial wave structure. By fixing position, you obtain a displacement versus time graph showing temporal oscillation. These graphical representations make the wave equation's behavior visually interpretable for analyzing wave parameters.