Two Points Half a Wavelength Apart

Q. For the travelling harmonic wave y(x, t) = 2.0 cos 2π (10t – 0.0080 x + 0.35) where x and y are in cm and t in s, what is the phase difference between oscillatory motion of two points separated by a distance of λ/2, in radians? expects 3.1 rad

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Q. For the travelling harmonic wave y(x, t) = 2.0 cos 2π (10t – 0.0080 x $k = 0.0080$ rev/cm + 0.35) where x and y are in cm and t in s, what is the phase difference between oscillatory motion of two points separated by a distance of λ/2 $Δ x = \frac{λ}{2}$ , in radians?

The playground

Outputs

phase

Δ φ

The solution

Solved with TravellingWave

Given

$k = 0.0080$ rev/cm from “2π (10t – 0.0080 x”
$Δ x = \frac{λ}{2}$ from “λ/2”

Find $Δ φ$ .

Working

$k = 0.0080$ rev/cm $= 5.02654825$ m⁻¹
wave number and wavelength(wave) $k = \frac{2 \cdot π}{λ}$
$5.02654825 = \frac{2 \cdot 3.14159265}{λ}$
$λ = 1.25$ m
λ/2 $Δ x = \frac{λ}{2}$
$Δ x = \frac{1.25}{2}$
$Δ x = 0.625$ m
phase across a separation(wave) $Δ φ = k \cdot Δ x$
$Δ φ = 5.02654825 \cdot 0.625$
$Δ φ = 3.142$ rad
$Δ φ \approx 3.1$ radto 2 significant figures

Conclusion

A whole wavelength is a full turn of phase, so half of one puts the two points in opposite phase, whatever the wave.

The Ruby

1require_relative "../physics"23question :two_points_half_a_wavelength_apart do4  kind :numerical5  describes "For the travelling harmonic wave y(x, t) = 2.0 cos 2π (10t – 0.0080 x + 0.35) where x and y are in cm " \6            "and t in s, %{phase} separated by a distance of λ/2, in radians?"7  asks phase: "what is the phase difference between oscillatory motion of two points"8end

1require_relative "../questions/two_points_half_a_wavelength_apart"2require_relative "../science/waves/travelling_wave"34solve :two_points_half_a_wavelength_apart, using: TravellingWave do5  objects :wave67  deduce_from "2π (10t – 0.0080 x", wave.wave_number => "0.0080 rev/cm"8  deduce_from "λ/2", wave.separation => wave.wavelength / 2910  phase = wave.phase_difference11  answer phase:, in: "rad", figures: 21213  conclusion "A whole wavelength is a full turn of phase, so half of one puts the two points in opposite phase, " \14             "whatever the wave."15end

1require_relative "../solutions/two_points_half_a_wavelength_apart"23play_with :two_points_half_a_wavelength_apart do4  title "Two Points Half a Wavelength Apart"5end

The laws

quantity	written	units	within	from
pi	$π$	—	0 … ∞	TravellingWave
amplitude	$a$	m	—	TravellingWave
wave number	$k$	m⁻¹	—	TravellingWave
angular frequency	$ω$	rad/s	—	TravellingWave
wavelength	$λ$	m	0 … ∞	TravellingWave
period	$T$	s	—	TravellingWave
frequency	$ν$	Hz	0 … ∞	TravellingWave
wave speed	$v$	m/s	—	TravellingWave
separation	$Δ x$	m	—	TravellingWave
phase difference	$Δ φ$	—	—	TravellingWave
wave number of ground	$k_{g}$	m⁻¹	—
wavelength of ground	$λ_{g}$	m	0 … ∞
wave number of wave	$k$	m⁻¹	—
wavelength of wave	$λ$	m	0 … ∞
angular frequency of ground	$ω_{g}$	rad/s	—
period of ground	$T_{g}$	s	—
angular frequency of wave	$ω$	rad/s	—
period of wave	$T$	s	—
frequency of ground	$ν_{g}$	Hz	0 … ∞
frequency of wave	$ν$	Hz	0 … ∞
wave speed of ground	$v_{g}$	m/s	—
wave speed of wave	$v$	m/s	—
phase difference of ground	$Δ φ_{g}$	—	—
separation of ground	$Δ x_{g}$	m	—
phase difference of wave	$Δ φ$	—	—
separation of wave	$Δ x$	m	—
amplitude of wave	$a$	m	—

k_{w} = \frac{2 \cdot π}{λ_{w}}

wave number and wavelength

ω_{w} = \frac{2 \cdot π}{T_{w}}

angular frequency and period

ν_{w} = \frac{1}{T_{w}}

frequency and period

v_{w} = ν_{w} \cdot λ_{w}

speed of a wave

Δ φ_{w} = k_{w} \cdot Δ x_{w}

phase across a separation

k_{g} = \frac{2 \cdot π}{λ_{g}}

wave number and wavelength(ground)

k = \frac{2 \cdot π}{λ}

wave number and wavelength(wave)

ω_{g} = \frac{2 \cdot π}{T_{g}}

angular frequency and period(ground)

ω = \frac{2 \cdot π}{T}

angular frequency and period(wave)

ν_{g} = \frac{1}{T_{g}}

frequency and period(ground)

ν = \frac{1}{T}

frequency and period(wave)

v_{g} = ν_{g} \cdot λ_{g}

speed of a wave(ground)

v = ν \cdot λ

speed of a wave(wave)

Δ φ_{g} = k_{g} \cdot Δ x_{g}

phase across a separation(ground)

Δ φ = k \cdot Δ x

phase across a separation(wave)

Δ x = \frac{λ}{2}

λ/2