When I was a high school student, I spent many hours practicing the trumpet. As evening would approach, and the sun went down, I could hear my notes echoing off a school building some 500 meters away. Back then, I often wondered why I could hear those faraway echoes only after sunset. This blog post will describe this curious phenomenon and simulate it using the COMSOL Multiphysics® software and its unique ray-tracing method.
Sound Refraction Due to Temperature Gradient
I have since come to learn that the cause of this sound phenomenon is the change in temperature distribution in the atmosphere. Temperature generally decreases with an increase in altitude. In this situation, the speed of sound in the air becomes slower with rising altitude due to the temperature dependency of the speed of sound. For example, the sound speed in air c_0 is well described by the following ideal gas model.
Here, \gamma, R_{\rm{const}}, T, and M_{\rm{n}} stand for the ration of specific heats, the universal gas constant, the temperature, and the molar mass, respectively. The ideal gas model assumes a dry air, but, generally speaking, the sound speed in air also depends on the relative humidity. For example, the calibration coupler model exemplifies the sound speed in moist air. According to Snell’s law, when an acoustic wave is incident on the interface from the slower sound speed region to the faster region, a refracted wave with a smaller refraction angle than incident angle occurs, as shown below.
A sound refraction on the interface between the slower and the faster sound speed region.
Thus, in the continuous graded temperature field under the standard atmosphere condition, sounds are refracted upward as shown in the following figure.
Example of sound ray propagation during the day. The lines represent the sound rays, and the background color represents the temperature field, with temperature warming toward the ground.
Sounds propagating in the atmosphere generally disappear into the sky. However, temperature inversion — when the heat radiation from the surface exceeds the received heat from the sun, such as at night — sometimes occurs. This inversion refracts sounds downward, as shown below.
Example of sound ray propagation at night.
As a result of the reversed direction of sound propagation, you can hear more distant sounds at night. The accurate modeling of this phenomenon is crucial for outdoor sound analysis because the difference in the sound transfer characteristics greatly affects the calculation results of factors such as outdoor noise level and speech intelligibility.
The ray-tracing methods are appropriate for simulating sound propagation in large outdoor spaces since they do not require a fine spatial mesh that resolves the wavelength, like the pressure- or wave-based methods. However, in the standard ray-tracing method, the ray directions only change when rays hit a boundary with a reflection or refraction condition. To calculate the smooth, refracted sound ray paths in the atmosphere, the multiple boundaries, each of which describes the refraction condition, must be set manually. This process can become time consuming, and often, it is unclear what the appropriate settings should be. The Hamiltonian based ray-tracing method, which is implemented in COMSOL Multiphysics®, is well-suited for outdoor sound analysis because it can model ray trajectories in graded media accurately and inherently.
To learn more about the capability of the ray-tracing algorithm used in COMSOL Multiphysics®, refer to our blog post “How Does the Choice of Ray-Tracing Algorithm Affect the Solution?” This method is also appropriate for modeling ocean acoustic problems such as in this tutorial model Underwater Ray Tracing in a 2D Axisymmetric Geometry, where the speed of sound depends on depth.
Below, we will use the Ray Acoustics interface to calculate the different sound ray trajectories occurring outdoors during the day and at night.
Modeling Long-Path Echoes
Let’s use COMSOL Multiphysics® to simulate the long-path echo of my trumpet from the building 500 m away. To confirm whether the echo is detectable only at night, the simulation was conducted using two temperature conditions.
A trumpet, like the one the blog author used to practice with in high school. Image by Ballista and licensed under CC BY-SA 4.0, via Wikimedia Commons.
The simulation consists of the following steps:
- Calculate a temperature field with the Heat Transfer in Fluids interface
- Calculate a radiation directivity of the trumpet bell with the Pressure Acoustics, Frequency Domain interface
- Calculate a sound traveling outdoors with the Ray Acoustics interface
Model geometry and the zoomed-in source region.
The figure above shows the model geometry. The school building is represented by the white, open rectangular space. The shape of the ground was created using elevation data.
The simulation assumes that the sound is emitted from the bell of the trumpet and that the player is located near the bell (the player location is used for calculating impulse responses). The figure also shows the zoomed-in source region, described in Step 2.
Step 1: Heat Transfer Analysis
In the first step, two temperature conditions were assigned to the ground: 25°C during the day and 9°C at night, while the sky’s temperature was kept at 19°C. Other boundaries, including the surfaces of the school building, were set as Thermal Insulation boundary conditions. The figure below shows the temperature fields during the day and at night.
Temperature fields during the day and at night.
During the day, you can see the standard temperature distribution in a vertical direction — the upper section is colder. At night, the condition shows the inversion — the upper section is hotter.
Step 2: Pressure Acoustics Analysis
The second step models the sound radiation directivity of the trumpet’s bell. In the example, only the directivity derived from the bell shape was considered; any losses were not modeled. The bell shape was modeled as an exponential horn with a cutoff frequency of 1200 Hz using the Interior Sound Hard Boundary (Wall) condition. The cross-sectional radius of the exponential horn, r, grows with the following equation:
where
f_{\rm{c}} is the cutoff frequency, and c is the speed of sound. The space variable is represented by x. Note that the above equation is a two-dimensional version (assuming a uniform thickness in the out-of-plane direction). In the actual three-dimensional exponential horn, the cross-section area grows with e^{mx}.
In this step, only the inner area of the circle was modeled using finite element method (FEM) with truncation by perfectly matched layers (PML). As an excitation, a Normal Acceleration boundary condition of 1 m/s2 was set to the inlet of the bell. To switch the temperature fields obtained in Step 1, the Parametric Sweep study step with switching parameter “Ns” was used. The following screenshots show the settings of the Frequency Domain study step and the Parametric Sweep study step.
The settings of the Frequency Domain study step (left) and the Parametric Sweep study step (right).
The radiation pattern of the exponential horn calculated in day conditions (left) and night conditions (right).
The above figure presents the radiation pattern of the bell in two temperature fields at 125 Hz, 1000 Hz, and 4000 Hz. The directivity becomes sharper with the increase of the frequency. The radiation intensity at high frequencies is smaller above the cutoff frequency. Also, you can see temperature fields have little effect on the directivities.
Step 3: Ray Acoustics Analysis
The third step performs the sound ray tracing under the temperature field and the bell radiation directivity obtained in Step 1 and Step 2, respectively. The ray-tracing method in COMSOL Multiphysics® easily calculates an impulse response in the graded media. Select Compute intensity and power in graded media from the Intensity Computation list in the Ray Acoustics interface, as shown below.
The settings of the Intensity Computation in the Ray Acoustics interface.
The Wall node’s Mixed diffuse and specular reflection condition was applied to the boundaries on the ground and the building, while the Wall node’s Disappear condition was assigned to the other boundaries to model a nonreflective boundary. Probability of specular reflection of 0.95 was set to all reflective boundaries. As for the sound absorption modeling of the ground, the following approximated Wilson’s impedance model was used with the assumption that the underground was thick.
Here, z_{\rm{n}} is the normalized surface impedance of the ground. The air density and the flow resistivity are represented by \rho and \sigma, respectively. The imaginary unit and the angular frequency are represented by j and \omega, respectively. In this simulation, the flow resistivity was set to 440 kPa s/m2. The normal incidence absorption coefficient of the ground is plotted below.
Normal incidence absorption coefficient of the ground.
The absorption coefficient of 0.05 was set to the surfaces of the school building. To take the radiation directivities of the bell into account, the Release from Boundary feature was used.
Setting for Release from Boundary feature.
The blog post “Full Acoustic Room Impulse Response of a Small Smart Speaker” is a good reference for learning more about the Release from Boundary feature. In this blog post, it is important to note that the Release from Pressure Field feature is also discussed, it represents a more automated way of setting up the source in 3D models, a manual setup is achieved here using the Release from Boundary feature. To use the results from the source computation study, the Ray Tracing study step and the Parametric Sweep study step were set up as follows.
The settings of the Ray Tracing study step (left) and the Parametric Sweep study step (right).
Ray-Tracing Results
The following images present the sound ray trajectories and the impulse response for the day condition. The impulse response is captured by the receiver, which is set at the player’s position (near the bell).
Ray trajectories at 500 Hz in the day condition; the ray color represents the ray propagation time.
Impulse response at the player’s position in the day condition.
For visibility, the rays whose number of reflections is 0 were excluded from the figure of the ray trajectories. You can see that some rays hit the building surface but disappeared into the sky and do not return to the player’s position. The same tendency was confirmed at other frequencies. The impulse response at the player’s position did not capture the echo from the building. Subsequently, the sound ray trajectories and the impulse response at the player’s position for the night condition are shown below.
Ray trajectories at 500 Hz in the night condition.
Impulse response at the player’s position in the night condition.
Unlike the day condition, many rays hit the building and then returned to the player’s location at night. The impulse response at the player’s position also includes the echo from the school building. These results imply that the school building received louder sounds at night. To confirm how much louder, the averaged sound pressure level (SPL) on the surface of the school building on the player’s side is illustrated below.
The averaged SPL on the surface of the school building on the player’s side.
Here, the bell’s radiation characteristic (cut off at 1200 Hz), modeled by coupling with FEM (wave-based method) largely contributed to the frequency characteristics of the above SPL plot. The accurate modeling of the sound path in the graded temperature field clearly shows that sounds propagated to the faraway building better at night compared to at day. The building surface received sound that was more than 5.5 dB louder at 125 Hz–2000 Hz in the night condition. This difference could be significant if the source is not the trumpet, but instead, for example, condenser units in a factory located on a noise-regulated land border.
Finally, I would like to use auralization to share my experience. In the following examples, you can hear the differences in sound at the player’s position between the day and night conditions. In the second example, you will hear the echo from the school building at night. For more about auralization, refer to the “Convolution and Auralization for Room Acoustics Analysis” blog post.
Auralized trumpet sound during the day.
Auralized trumpet sound at night.
The Importance of Regulating and Controlling Sounds
The blog post has explained and modeled the well-known phenomenon that sounds travel farther at night. It has also demonstrated how the ray-tracing algorithm available in COMSOL Multiphysics® is uniquely suited to modeling large outdoor sound fields and appropriate for simulating the sound refraction derived from the graded media. Noise regulation usually demands lower sound levels at night than during the day, so it is important to consider the refraction characteristic of sounds in the atmosphere at night. The Ray Acoustics interface in COMSOL Multiphysics® can be used to reliably predict and control outdoor sounds, as well as evaluate the speech intelligibility of outdoor public address systems, which often necessitate high quality, such as when addressing large crowds.
The elevation data of the ground was created from the color elevation map provided by Geospatial Information Authority of Japan.
The anechoic sound is provided by The Open AIR Library under CC BY 4.0.
Comments (0)