{"product_id":"levitah-held-by-sound-physics-t-shirt","title":"Levitah – Held by Sound Physics T-Shirt","description":"\u003cp\u003eLevitah holds a row of pearls still in mid-air, caught in a column of sound no one can see.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eSound carries momentum, and a strong standing wave exerts a steady average force, the acoustic radiation force, on small objects. King computed it for spheres in 1934. Gor'kov (1962) wrote it, for a sphere much smaller than the wavelength in an ideal fluid, as the gradient of a potential U, with a compressibility factor f₁ and a density factor f₂.\u003c\/p\u003e\u003cp\u003eFor a small bead much denser and much less compressible than air (ρ_p ≫ ρ₀, κ_p ≪ κ₀), both factors are close to 1. U is then lowest where the pressure amplitude vanishes and the velocity amplitude peaks, so the bead collects near the pressure nodes, which are the velocity antinodes, half a wavelength apart. At 40 kHz in air the wavelength is about 8.6 mm, so the trapping sites are about 4.3 mm apart.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eU = (4π\/3) a³ [ f₁ · ½ κ₀ ⟨p²⟩ − f₂ · ¾ ρ₀ ⟨v²⟩ ],   F = −∇U\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eU: radiation potential (J)\u003c\/li\u003e\n\u003cli\u003ea: sphere radius, a ≪ wavelength (m)\u003c\/li\u003e\n\u003cli\u003ef₁ = 1 − κ_p\/κ₀: compressibility contrast\u003c\/li\u003e\n\u003cli\u003ef₂ = 2(ρ_p − ρ₀)\/(2ρ_p + ρ₀): density contrast\u003c\/li\u003e\n\u003cli\u003eκ₀, ρ₀: compressibility and density of the air\u003c\/li\u003e\n\u003cli\u003e⟨p²⟩, ⟨v²⟩: time-averaged squared acoustic pressure and velocity\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eKing, Proc. R. Soc. Lond. A 147, 212 (1934); Gor'kov, Sov. Phys. Dokl. 6, 773 (1962).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e It is not anti-gravity, and it does not work on any object: size and density limits apply. Dense, rigid beads collect near pressure nodes (velocity antinodes), not at velocity nodes; in a vertical levitator gravity holds each bead slightly below its node.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Levitah, the pixel-art character, with “HELD BY SOUND”. Back: Gor'kov's potential, the force, the pressure-node rule, the 40 kHz spacing and the 1934 and 1962 sources. Printed on a unisex tee.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"S \/ White","offer_id":50960522739815,"sku":"21046557942677456653","price":29.99,"currency_code":"USD","in_stock":true},{"title":"S \/ Black","offer_id":50960522772583,"sku":"33398304029196882871","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ White","offer_id":50960522805351,"sku":"18661394125957292655","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ Black","offer_id":50960522838119,"sku":"57721287095030250953","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ White","offer_id":50960522870887,"sku":"26002821482747166592","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ Black","offer_id":50960522903655,"sku":"24208854558035946088","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ White","offer_id":50960522936423,"sku":"10261556108572334958","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ Black","offer_id":50960522969191,"sku":"11512871089282608593","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ White","offer_id":50960523001959,"sku":"13359762500350173417","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ Black","offer_id":50960523034727,"sku":"14941788106412729733","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ White","offer_id":50960523067495,"sku":"31425531214151502156","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ Black","offer_id":50960523100263,"sku":"69703523808046265801","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ White","offer_id":50960523133031,"sku":"33119361773357305964","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ Black","offer_id":50960523165799,"sku":"47060016471621317589","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ White","offer_id":50960523198567,"sku":"25425330725156332335","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ Black","offer_id":50960523231335,"sku":"22715876481073907060","price":29.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/acoustic-levitation_front_01_white.png?v=1793625877","url":"https:\/\/pixelatedphysics.com\/ko\/products\/levitah-held-by-sound-physics-t-shirt","provider":"Pixelated Physics","version":"1.0","type":"link"}